Involutions on Generating Functions

Masanari Kida Department of Mathematics

University of Electro-Communications

Chofu, Tokyo 182-8585

Japan

masanari.kida@gmail.com

Yuichiro Urata

NTT Network Technology Laboratories

NTT Corporation 3-9-11 Midori-cho, Musashino-shi

Tokyo 180-8585

Japan

urata.yuichiro@lab.ntt.co.jp

Abstract

We study a family of involutions on the space of sequences. Many arithmetically or combinatorially interesting sequences appear as eigensequences of the involutions. We develop new tools for studying sequences using these involutions.

1 Introduction

In his paper [8], Kaneko proved an interesting identity

∑_{i=0}^{n+1} \binom{n + 1}{i}(n + i + 1)B_{n+i} = 0  (1)

for Bernoulli numbers B_{n} defined by the exponential generating function

\frac{x}{e^{x} − 1} = ∑_{n=0}^{∞} B_{n} \frac{x^{n}}{n!} .  (2)

By using Kaneko’s identity, we need about half the number of terms for computing B_{n} compared with the usual identity

∑_{i=0}^{n} \binom{n + 1}{i}(−1)^{i}B_{i} = n + 1.

Kaneko proved the identity by means of a continued fraction expansion. His paper also contains a sketch of another proof due to Zagier who uses an involutive linear action on sequences. Zagier’s involution was further studied by various authors (see, for example, [6, 12, 13]).

The aim of this paper is to define an infinite family of involutions on the space of sequences, which generalize Zagier’s involution, and to prove various properties of the involutions and their eigensequences. The definition of the involutions looks like that of modular forms and this resemblance enables us to show that generating functions of eigensequences enjoy properties analogous to modular forms. The importance of these involutions is not only to provide a tool to study general sequences but also to elicit a symmetry in classical and important sequences appearing as eigensequences. In fact, as an application, we show that these involutions yield many interesting identities involving the Bernoulli numbers, the Fibonacci numbers and so on.

This paper is organized as follows. In the next section, we define the involutions and show their basic properties. In Section 3, we give examples of eigensequences of the involutions. It will become apparent that many arithmetically and/or combinatorially interesting sequences (namely, many core sequences in OEIS [15]) are eigensequences. In Section 4, we construct differential operators on the generating functions of eigensequences. In particular, we prove the existence of analogues of the Cohen-Rankin brackets. These operators produce new eigensequences from given eigensequences. In Section 5, we consider the action of the involutions on the endomorphism ring of the sequences. By studying this action, we can produce infinitely many linear identities for eigensequences in Section 6 including a generalization of Kaneko’s identity.

Throughout this paper, the binomial coefficient is a generalized one defined by

\binom{n}{k} = \frac{n(n − 1) ⋯ (n − k + 1)}{k!} = \frac{n\underline{k}}{k!}

for k ≥ 0, where n\underline{k} is the falling factorial power defined by

n\underline{k} = n(n − 1) ⋯ (n − k + 1).

2 Definition and basic properties of involutions

Let 𝒮 be the set of sequences in a field F of characteristic 0. Though we are primarily interested in integer or rational sequences, we do not need such restriction to develop a general theory. Thus we start with a general field F.

Since we have to deal with many sequences and their generating functions, we need a more systematic notation than usual. We denote the geometric series {r^{n}} of the ratio r by ⟨r⟩ and also by δ_{n} the sequence whose terms are all 0 except for the n-th term which is 1. For any sequence a = {a_{n}} ∈ 𝒮, we associate formal power series

G_{i}(a, x) = ∑_{n=0}^{∞} a_{n} \frac{x^{n}}{(n!)^{i}} ∈ F [[x]] (i = 0, 1).

The series G_{0}(a, x) is the ordinary generating function and G_{1}(a, x) is the exponential generating function of a. Further, we use the notation

[n]a = a_{n} and [\frac{x^{n}}{(n!)^{i}} ] G_{i}(a, x) = a_{n}.

By the correspondence a ↦ G_{i}(a, x), we have isomorphisms of F -vector spaces:

φ_{i} : 𝒮 ⟶ F [[x]].

By these isomorphisms, we can endow 𝒮 with F -algebra structures. Namely we can define products on 𝒮 by

[n](a ∗_{i} b) = ∑_{j=0}^{n} \binom{n}{j}^{i}a_{j}b_{n−j}.

The operation ∗_{0} is called the ordinary convolution and ∗_{1} is the binomial convolution. We set 𝒮_{i} = φ_{i}(𝒮).

In addition, we denote the term-wise product sequence simply by ab, that is, [n](ab) = [n]a · [n]b.

Definition 1. We define an action of a lower triangular 2 × 2 regular matrix A = [\begin{matrix}a & 0 \\ c & d \end{matrix}]

over F on 𝒮_{0} and 𝒮 by

G_{0}|_{[A]_{k}}(a, x) = G_{0}(a|_{[A]_{k}}, x) = (cx + d)^{−k}G_{0} (a, \frac{ax}{cx + d}) ,

where k is an integer called the weight of the action.

It is plain to see that this operation satisfies

(G_{0}|_{[A]_{k}})|_{[B]_{k}} = G_{0}|_{[AB]_{k}}  (3)

for any such matrices A and B. Thus this is a well-defined action on 𝒮. The n-th term of the new sequence is explicitly given by

[n](a|_{[A]_{k}}) = \frac{1}{d^{n+k}} ∑_{j=0}^{n} \binom{n + k − 1}{j} a_{n−j} a^{n−j} (−c)^{j}.  (4)

If the order of A in GL_{2}(F) is 2, then we have an involution on 𝒮. Therefore we are particularly interested in the actions of the following matrices of order 2:

−I = [\begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix}] , A_{c} = [\begin{matrix}−1 & 0 \\ c & 1 \end{matrix}] , and − A_{c} = [\begin{matrix}1 & 0 \\ −c & −1 \end{matrix}] ,

where c is any element in F. (Later at some places we take c from a ring containing F). For these matrices, the actions are given by

[n](a|_{[−I]_{k}}) = (−1)^{k}a_{n},
[n](a|_{[A_{c}]_{k}}) = (−1)^{n} ∑_{j=0}^{n} \binom{n + k − 1}{j} a_{n−j} c^{j},  (5)

[n](a|_{[−A_{c}]_{k}}) = (−1)^{n+k} ∑_{j=0}^{n} \binom{n + k − 1}{j} a_{n−j} c^{j}.  (6)

In particular, a special case

[n](a|_{[A_{−1}]_{1}}) = ∑_{j=0}^{n} \binom{n}{j} (−1)^{j}a_{j}

is the action which Zagier [8] used. Since the action of −I is more or less trivial, we will not consider it further and we assume that A is one of ±A_{c} unless otherwise specified. We write down the first few terms of a|_{[A_{c}]_{k}} for later convenience:

\begin{aligned}[0](a|_{[A_{c}]_{k}}) & = a_{0}; \\ [1](a|_{[A_{c}]_{k}}) & = −ka_{0}c − a_{1}; \\ [2](a|_{[A_{c}]_{k}}) & = \binom{1 + k}{2} a_{0}c^{2} + (1 + k)a_{1}c + a_{2}; \\ [3](a|_{[A_{c}]_{k}}) & = −\binom{2 + k}{3}a_{0}c^{3} − \binom{2 + k}{2}a_{1}c^{2} − (2 + k)a_{2}c − a_{3}. \end{aligned}  (7)

The action of [A]_{k} on 𝒮 induces an action on 𝒮_{1}.

Proposition 2. Let a ∈ 𝒮. For positive integers k, we have

G_{1}(a|_{[A]_{k}}, x) = \frac{1}{d^{k}} \frac{d^{k−1}}{dx^{k−1}} (exp (−\frac{c}{d}x) \underbrace{∫ ⋯ ∫}_{k−1} G_{1} (⟨\frac{a}{d}⟩ a, x) \underbrace{dx ⋯ dx}_{k−1}) ,

where all the integral sign means the formal integration from 0 to x.

Proof. We compute

G_{1}(a|_{[A]_{k}}, x) = \frac{1}{d^{k}} ∑_{n≥0} \frac{1}{d^{n}} (∑_{j=0}^{n} \binom{n + k − 1}{j}(−c)^{j}a^{n−j}a_{n−j}) \frac{x^{n}}{n!}

= \frac{1}{d^{k}} ∑_{n≥0} \frac{(n + k − 1)!}{n!} (∑_{j=0}^{n} \binom{n}{j} (−\frac{c}{d})^{j} (\frac{a}{d})^{n−j} a_{n−j}(n − j)!\frac{1}{(n − j + k − 1)!}) \frac{x^{n}}{n!} .

The inner sum is the binomial convolution of

⟨−\frac{c}{d}⟩ and (\frac{a}{d})^{n} a_{n} \frac{n!}{(n + k − 1)!}.

The exponential generating function of the latter sequence is

x^{1−k} \underbrace{∫ ⋯ ∫}_{k−1} G_{1} (⟨\frac{a}{d}⟩ a, x) \underbrace{dx ⋯ dx}_{k−1} .

This completes the proof of the proposition.

The special case where k = 1 will be important for us:

G_{1}(a|_{[A]_{1}}, x) = \frac{1}{d} exp (−\frac{c}{d}x) G_{1} (⟨\frac{a}{d}⟩ a, x) .  (8)

Remark 3. We can prove the following formula similarly for k ≤ 0:

[\frac{x^{n}}{n!} ] G_{1}(a|_{[A]_{k}}, x) = [\frac{x^{n}}{n!} ] \frac{1}{d^{k}} \underbrace{∫ ⋯ ∫}_{1−k} (exp (−\frac{c}{d}x) \frac{d^{1−k}}{dx^{1−k}} G_{1} (⟨\frac{a}{d}⟩ a, x)) \underbrace{dx ⋯ dx}_{1−k}

provided n ≥ 1 − k.

Now we define the main object of our study.

Definition 4. Let A be a lower triangular regular matrix of order 2. A sequence a satisfying

a|_{[A]_{k}} = sa

with some s ∈ { ±1} is called an eigensequence of [A]_{k}. We call s the sign of the eigensequence. We denote the eigenspaces by

𝒮 (A)_{k}^{+} = {a ∈ 𝒮 : a|_{[A]_{k}} = +a} and 𝒮 (A)_{k}^{−} = {a ∈ 𝒮 : a|_{[A]_{k}} = −a} .

Although a resemblance between this definition and that of modular forms is clear (see [11] for example), several differences will appear in the following. Here we only note that each space 𝒮 (A)_{k}^{±} is an infinite-dimensional F -vector space for every integer k.

By (4) we have

a ∈ 𝒮 (A)_{k}^{±} ⇔ (±1 − \frac{a^{n}}{d^{n+k}} ) a_{n} = \frac{1}{d^{n+k}} ∑_{j=1}^{n} \binom{n + k − 1}{j}a_{n−j}a^{n−j}(−c)^{j}.  (9)

If A = ±A_{c}, then (±1 − \frac{a^{n}}{d^{n+k}} ) equals ±2 or 0 alternatively with n. This means that we can choose one of the two consecutive terms freely and the next term is determined automatically by the preceding terms. We also define

𝒮_{i}(A)_{k}^{±} = φ_{i}(𝒮 (A)_{k}^{±}).

When A = A_{0}, the eigensequences are easy to describe.

Proposition 5. The following equivalences hold for i = 0, 1:

a ∈ 𝒮 (A_{0})_{k}^{+} ⟺ G_{i}(a, x) is an even power series;
a ∈ 𝒮 (A_{0})_{k}^{−} ⟺ G_{i}(a, x) is an odd power series.

Proof. If A = A_{0}, then Definition 1 implies that a ∈ 𝒮 (A_{0})_{k}^{±} if and only if

G_{0}(a, −x) = ±G_{0}(a, x).

This means that G_{0}(a, x) is an odd or even power series according to the sign of a. Moreover, G_{0}(a, x) is an odd (resp. even) power series if and only if G_{1}(a, x) is an odd (resp. even) power series. This completes the proof. Example 6. The sequence of the Euler numbers E = {E_{n}} (see [7, p.559]) is an example in 𝒮 (A_{0})_{k}^{+}, because it is defined by

G_{1}(E, x) = sec x = ∑_{n=0}^{∞} E_{2n} \frac{x^{2n}}{(2n)!}.  (10)

An example from 𝒮 (A_{0})_{k}^{−} is the sequence of the tangent numbers T = {T_{n}} defined by

G_{1}(T , x) = tan x = ∑_{n=0}^{∞} T_{2n+1} \frac{x^{2n+1}}{(2n + 1)!},

which have a close relation to the Bernoulli numbers [7, p.287].

Also we can deduce the following identities readily from (5) and (6). Proposition 7. We have

𝒮 (A_{c})_{k}^{±} = \begin{cases}𝒮 (−A_{c})_{k}^{±}, & if k is even; \\ 𝒮 (−A_{c})_{k}^{∓}, & if k is odd. \end{cases}

By these propositions, we shall mainly consider the eigenspaces 𝒮 (A_{c})_{k}^{±} with c ≠ 0. As in the case of modular forms, the space of eigensequences has a graded structure.

Proposition 8. If a ∈ 𝒮 (A)_{k_{1}}^{+} and b ∈ 𝒮 (A)_{k_{2}}^{+}, then a ∗_{0} b ∈ 𝒮 (A)_{k_{1}+k_{2}}^{+}. If A ≠ ±A_{0}, then 𝒮 (A)^{+} = ⊕_{k∈ℤ}𝒮 (A)_{k}^{+} is a graded ring under this product. Proof. Let a ∈ 𝒮 (A)_{k_{1}}^{+} and b ∈ 𝒮 (A)_{k_{2}}^{+}. Then we have

G_{0}|_{[A]_{k_{1}+k_{2}}}(a ∗_{0} b, x) = (G_{0}(a, x)G_{0}(b, x))|_{[A]_{k_{1}+k_{2}}}

= (cx + d)^{−(k_{1}+k_{2})}G_{0} (a, \frac{ax}{cx + d}) G_{0} (b, \frac{ax}{cx + d})

= G_{0}(a|_{[A]_{k_{1}}}, x)G_{0}(b|_{[A]_{k_{2}}}, x)

= G_{0}(a, x)G_{0}(b, x)

= G_{0}(a ∗_{0} b, x).

This shows the first assertion. To show the second assertion, it suffices to prove that if k_{1} ≠ k_{2}, then 𝒮 (A)_{k_{1}}^{+} ∩ 𝒮 (A)_{k_{2}}^{+} = {0}. Let a ∈ 𝒮 (A)_{k_{1}}^{+} ∩ 𝒮 (A)_{k_{2}}^{+}. Then

(cx + d)^{k_{1}}G_{0} (a, \frac{ax}{cx + d}) = (cx + d)^{k_{2}}G_{0} (a, \frac{ax}{cx + d})

holds. Since F [[x]] is an integral domain, we have k_{1} = k_{2} provided c ≠ 0.

In a similar manner, we can show the following proposition.

Proposition 9. Let s_{1}, s_{2} ∈ {±1}. If a ∈ 𝒮 (A)_{k_{1}}^{s_{1}} and b ∈ 𝒮 (A)_{k_{2}}^{s_{2}}, then a∗_{0}b ∈ 𝒮 (A)_{k_{1}+k_{2}}^{s_{1}s_{2}}.

By these propositions, it is clear that considering these involutions with varying k rather than a fixed k.

A special case of Proposition 9 implies the following.

Proposition 10. There exists an isomorphism

𝒮 (A_{c})_{k}^{+} ≅ 𝒮 (A_{c})_{k−1}^{−}, a ↦ δ_{1} ∗_{0} a

of vector spaces for every integer k.

Proof. Since G_{0}(δ_{1}, x) = x, we can see δ_{1} ∈ 𝒮 (A_{c})_{−1}^{−}. It follows from Proposition 9 that the convolution δ_{1} ∗_{0} a belongs to 𝒮 (A_{c})_{k−1}^{−}. The given map is obviously an injective linear map. To show the surjectivity, we note that the sequence δ_{1} ∗_{0} a is a shift of a to the

right. Therefore it is enough to prove that every sequence in 𝒮 (A_{c})_{k−1}^{−} begins with 0. This assertion follows immediately from (7).

In terms of generating functions, Proposition 10 gives the following isomorphism:

𝒮_{0}(A)_{k}^{+} ≅ 𝒮_{0}(A)_{k−1}^{−}, G_{0}(a, x) ↦ xG_{0}(a, x).

It is worth while to remark that the same correspondence gives an injective linear map from 𝒮 (A_{c})_{k}^{−} to 𝒮 (A_{c})_{k−1}^{+} but this is not surjective in general.

There is a variant of Proposition 9 for binomial convolution. Before we state it, we note the following equivalence. If k = 1, then by (8) we see

a ∈ 𝒮 (A_{c})_{1}^{±} ⟺ G_{1}(a, x) = ± exp(−cx)G_{1}(⟨−1⟩a, x),  (11)

which is a generalization of [12, Theorem 3.2]. Observe here that we have an identity

G_{1}(⟨−1⟩a, x) = G_{1}(a, −x).

Proposition 11. Let s_{1}, s_{2} ∈ {±1}. If a ∈ 𝒮 (A_{c_{1}})_{1}^{s_{1}} and b ∈ 𝒮 (A_{c_{2}})_{1}^{s_{2}}, then

a ∗_{1} b ∈ 𝒮 (A_{c_{1}+c_{2}})_{1}^{s_{1}s_{2}}.

Proof. By (11), we have

G_{1}(a, x) = s_{1} exp(−c_{1}x)G_{1}(⟨−1⟩a, x),
G_{1}(b, x) = s_{2} exp(−c_{2}x)G_{1}(⟨−1⟩b, x).

Multiplying the both sides, we obtain

G_{1}(a, x)G_{1}(b, x) = s_{1}s_{2} exp(−(c_{1} + c_{2})x)G_{1}(⟨−1⟩a, x)G_{1}(⟨−1⟩a, x).

The left hand side is equal to G_{1}(a ∗_{1} b, x). From the right hand side, we get

G_{1}(⟨−1⟩a, x)G_{1}(⟨−1⟩a, x) = G_{1}(a, −x)G_{1}(b, −x) = G_{1}(a ∗_{1} b, −x) = G_{1}(⟨−1⟩(a ∗_{1} b), x).

Now the proposition follows from (11).

Both Propositions 9 and 11 will be generalized in a broader context in Section 4 (see Theorems 31 and 33).

If the 0-th term of a sequence a ∈ 𝒮 is not 0, then G_{i}(a, x) (i = 0, 1) are invertible in F [[x]].

Corollary 12. If a ∈ 𝒮 (A_{c})_{k}^{+} and [0]a ≠ 0, then we have

G_{0}(a, x)^{−1} ∈ 𝒮_{0}(A_{c})_{−k}^{+}.

Moreover, if k = 1, then we have

G_{1}(a, x)^{−1} ∈ 𝒮_{1}(A_{−c})_{1}^{+}.

Proof. If the generating functions are invertible, then G_{i}(a, x)G_{i}(a, x)^{−1} = 1 = G_{i}(δ_{0}, x). Since δ_{0} belongs to 𝒮 (A_{c})_{0}^{+} ∩𝒮 (A_{0})_{k}^{+} for all c and k, the corollary follows from Propositions 9 and 11.

The following propositions give easy ways to alter c.

Proposition 13. If a ∈ 𝒮 (A_{c})_{k}^{±}, then ⟨r⟩a ∈ 𝒮 (A_{cr})_{k}^{±}.

We omit the proof since it is straightforward.

Proposition 14. If a ∈ 𝒮 (A_{c_{1}})_{k}^{±}, then a|_{[A_{c_{2}}]_{k}} ∈ 𝒮 (A_{2c_{2}−c_{1}})_{k}^{±}.

Proof. Using (3) and the relation A_{c_{2}}A_{2c_{2}−c_{1}} = A_{c_{1}}A_{c_{2}}, we compute

(a|_{[A_{c_{2}}]_{k}})_{[A_{2c_{2}−c_{1}}]_{k}} = a|_{[A_{c_{2}}A_{2c_{2}−c_{1}}]_{k}} = a|_{[A_{c_{1}}A_{c_{2}}]_{k}} = (a|_{[A_{c_{1}}]_{k}})_{[A_{c_{2}}]_{k}} = ±a|_{[A_{c_{2}}]_{k}}.

This proves the assertion.

We close this section with the following remark. For each A of order 2 and each k, we have projections

π(A)_{k}^{+} = \frac{1 + [A]_{k}}{2} : 𝒮 ⟶ 𝒮 (A)_{k}^{+}

and

π(A)_{k}^{−} = \frac{1 − [A]_{k}}{2} : 𝒮 ⟶ 𝒮 (A)_{k}^{−}.

Also by Proposition 8, if A ≠ ±A_{0}, then there is a surjection

π(A) = ⨁_{k∈ℤ} π(A)_{k}^{+} : 𝒮 ⟶ 𝒮 (A)^{+} = ⨁_{k∈ℤ} 𝒮 (A)_{k}^{+}.

The kernel of π(A) is ∩_{k}𝒮 (A)_{k}^{−}. In a similar manner as in the proof of Proposition 8, it is shown that it is actually 0. Hence the surjection is an isomorphism.

3 Eigensequences

In this section, we give various examples of sequences in 𝒮 (A_{c})_{k}^{±} (see Propositions 5 and 7). Although it is easy to construct such sequences from arbitrary sequences using the projections π(A)_{k}^{±}, almost all sequences thus obtained are unnatural and useless. Therefore we systematically search interesting eigensequences in the following. In the course of searching, we can reveal the structure of 𝒮 (A_{c})_{k}^{±} (Theorem 18). Note that Z.-H. Sun [12, Section 2] found some examples in 𝒮 (A_{−1})_{1}^{+} and some of the following examples are generalizations of his.

We begin with necessary conditions for a given sequence to be an eigensequence.

Lemma 15. Let a ∈ 𝒮.

1. If a ∈ 𝒮 (A_{c})_{k}^{+}, then

a_{1} = −\frac{1}{2}kca_{0},  (12)
a_{3} = \frac{1}{24}c(2 + k)(−12a_{2} + a_{0}c^{2}k + a_{0}c^{2}k^{2}).

Moreover k is an integer solution of the quadratic equation

(a_{1}^{3} + 3a_{0}^{2}a_{3} − 3a_{0}a_{1}a_{2})k^{2} + (3a_{1}^{3} − 6a_{0}a_{1}a_{2})k + 2a_{1}^{3} = 0.  (13)

2. If a ∈ 𝒮 (A_{c})_{k}^{−}, then

a_{0} = 0,  (14)
a_{2} = −\frac{1}{2}(1 + k)ca_{1},  (15)
a_{4} = \frac{1}{24}c(3 + k)(−12a_{3} + a_{1}c^{2}(k + 1) + a_{1}c^{2}(k + 1)^{2}).  (16)

Moreover k is an integer solution of the quadratic equation

(a_{2}^{3} + 3a_{1}^{2}a_{4} − 3a_{1}a_{2}a_{3})k^{2} + (6a_{1}^{2}a_{4} − 12a_{1}a_{2}a_{3} + 5a_{2}^{3})k + 3a_{1}^{2}a_{4} + 6a_{2}^{3} − 9a_{1}a_{2}a_{3} = 0.

Proof. The conditions on each term is obtained directly from the identities following (7) (see also (9)). By eliminating c from these equalities, we obtain the equations for k.

Note that the latter half 2 also follows from the isomorphism

𝒮 (A_{c})_{k+1}^{+} ≅ 𝒮 (A_{c})_{k}^{−}

in Proposition 10. In other words, the relations (14), (15) and (16) can be obtained by a right shift of the relations in 1 and a substitution of k by k + 1.

Using this lemma, we can decide that a given sequence is not an eigensequence or determine candidates for c and k.

We first give some examples of eigensequences with polynomial generating functions, namely finite eigensequences.

Example 16 (Polynomial generating functions). If the generating function G_{0}(a, x) for a ∈ 𝒮 (A_{c})_{k}^{±} (c ≠ 0) is a polynomial of degree m ≥ 1, then it is easy to see that the weight k should be negative and equal or less than −m. If k is exactly equal to −m, then we can show that [A_{c}]_{−m} defines an involution on the subspace

F [x]_{m} = {G(x) ∈ F [x] : deg G(x) ≤ m}.

By Lemma 15, we have an inequality

dim_{F} (𝒮_{0}(A_{c})_{k}^{+} ∩ F [x]_{m}) ≥ dim_{F} (𝒮_{0}(A_{c})_{k}^{−} ∩ F [x]_{m}).

This implies

dim_{F} (𝒮_{0}(A_{c})_{k}^{+} ∩ F [x]_{m}) = ⌈\frac{m + 1}{2}⌉ , dim_{F} (𝒮_{0}(A_{c})_{k}^{−} ∩ F [x]_{m}) = ⌊\frac{m + 1}{2}⌋ .

As k gets smaller, we have additional constraints on the terms of eigensequences. Therefore the dimension gets smaller, too.

Suppose that we want to find generating functions of degree 1 for plus eigensequences. Setting a_{2} = a_{3} = 0 in (13), we have

a_{1}^{3}k^{2} + 3a_{1}^{3}k + 2a_{1}^{3} = 0.

This equation yields that k = −1 or −2. From (12) it follows that a_{1} = ca_{0}/2 or ca_{0}. We can easily show that these are indeed eigensequences. Similar calculation leads to the following polynomial generating functions:

deg G_{0} = 1

1 + \frac{c}{2}x ∈ 𝒮_{0}(A_{c})_{−1}^{+},  x ∈ 𝒮_{0}(A_{c})_{−1}^{−} ∩ 𝒮_{0}(A_{0})_{k}^{−},
1 + cx ∈ 𝒮_{0}(A_{c})_{−2}^{+};

deg G_{0} = 2

t + ctx + sx^{2} ∈ 𝒮_{0}(A_{c})_{−2}^{+},  x (1 + \frac{c}{2}x) ∈ 𝒮_{0}(A_{c})_{−2}^{−},
(1 + cx) (1 + \frac{c}{2}x) ∈ 𝒮_{0}(A_{c})_{−3}^{+},  x(1 + cx) ∈ 𝒮_{0}(A_{c})_{−3}^{−}
(1 + cx)^{2} ∈ 𝒮_{0}(A_{c})_{−4}^{+};

where s and t are arbitrary parameters.

Since δ_{1} ∈ 𝒮 (A_{c})_{−1}^{−}, we can prove that

G_{0}(δ_{m}, x) = x^{m} = G_{0}(δ_{1}, x)^{m} ∈ 𝒮_{0}(A_{c})_{−m}^{(−1)^{m}}

for positive integer m by using Proposition 9 repeatedly. Combining this with Proposition 5, we conclude

δ_{m} ∈ 𝒮 (A_{c})_{−m}^{(−1)^{m}} ∩ 𝒮 (A_{0})_{k}^{(−1)^{m}}  (17)

for all c and k.

These finite eigensequences themselves are not so interesting, but they provide primary ingredients of general eigensequences. We shall see this in the following rational generating functions.

Example 17 (Rational generating functions). Combining polynomial generating functions in various ways, we have rational generating functions of eigensequences, whose weights can be computed using Proposition 9 and Corollary 12. The simplest examples are

G_{0}(⟨c⟩, x) = \frac{1}{1 − cx} ∈ 𝒮_{0}(A_{−2c})_{1}^{+} ∩ 𝒮_{0}(A_{−c})_{2}^{+}.

We also have

G_{0}(a, x) = \frac{2 + cx}{t + tcx + sx^{2}} ∈ 𝒮_{0}(A_{c})_{1}^{+}, G_{0}(b, x) = \frac{x}{t + tcx + sx^{2}} ∈ 𝒮_{0}(A_{c})_{1}^{−}.

As a special case of a, we have the Lucas sequence L = {L_{n}} defined by the recurrence

L_{0} = 2, L_{1} = 1, L_{n} = L_{n−1} + L_{n−2}.

In fact, since G_{0}(L, x) = \frac{2 − x}{1 − x − x^{2}} , we have L ∈ 𝒮 (A_{−1})_{1}^{+}. As for b, we have the Fibonacci

sequence F = {F_{n}} defined by

F_{0} = 0, F_{1} = 1, F_{n} = F_{n−1} + F_{n−2},

whose generating function is G_{0}(F , x) = \frac{x}{1 − x − x^{2}} ∈ 𝒮_{0}(A_{−1})_{1}^{−}. This means that F and L are not only a basis of the 2-dimensional vector space of sequences satisfying the recurrence a_{n+1} = a_{n} + a_{n−1}, but also a basis as a ⟨[A_{−1}]_{1}⟩-module. This fact plays an essential role in the researches of Fibonacci numbers although it is not noticed explicitly.

The binomial coefficients \{\binom{n + k − 1}{n}\}_{n≥0} are also of this type, since the ordinary

generating function is (1 − x)^{−k}. Therefore we see

\{\binom{n + k − 1}{n}\}_{n≥0} ∈ 𝒮 (A_{−1})_{2k}^{−}.

The generating function

ℰ_{k}(x) = \frac{1}{(1 + \frac{c}{2}x)^{k}} = ∑_{n=0}^{∞} \binom{n + k − 1}{n} (−\frac{c}{2})^{n} x^{n} ∈ 𝒮_{0}(A_{c})_{k}^{+}

plays a special role.

Theorem 18. We have a direct sum decomposition

𝒮_{0}(A_{c})_{k}^{+} = F · ℰ_{k}(x) ⊕ x^{2}𝒮_{0}(A_{c})_{k+2}^{+}

for any integer k. Moreover, any generating function G_{0}(a, x) ∈ 𝒮_{0}(A_{c})_{k}^{+} can be written uniquely as a sum of the form

G_{0}(a, x) = ∑_{i=0}^{∞} α_{2i}x^{2i}ℰ_{k+2i}(x) (α_{2i} ∈ F).  (18)

If G_{0}(a, x) ∈ 𝒮_{0}(A_{c})_{k}^{−}, then it can be written uniquely as

G_{0}(a, x) = ∑_{i=0}^{∞} α_{2i+1}x^{2i+1}ℰ_{k+2i+1}(x) (α_{2i+1} ∈ F).  (19)

Proof. Consider the map g : 𝒮_{0}(A_{c})_{k}^{+} ⟶ F defined by G_{0}(x) ↦ G_{0}(0). The dimension of the image of g is at most 1. Since [x^{0}]ℰ_{k}(x) ≠ 0, the dimension is 1, indeed. Suppose that G_{0}(x) ∈ ker g. Then [x]G_{0}(x) is also 0 by (12). We obtain a factorization G_{0}(x) = x^{2}H(x). From (17), we have H(x) ∈ 𝒮_{0}(A_{c})_{k+2}^{+}. This proves the first assertion. Unless H(x) is zero or a constant, we can continue the same argument for H(x), we can expand G_{0}(x) like (18). If H(x) is zero or a constant, the process stops and obtain a finite sum expansion. The equation (19) follows from (18) by using the isomorphism in Proposition 10.

An explicit formula for α_{i} in (18) and (19) will be given later (Proposition 28).

This theorem provides more precise test for eigensequences than Lemma 15. But we should note that it is still difficult to extract interesting sequences from this structure theorem. That being so, we continue our quest.

Example 19. In this example, we deal with irrational generating functions. Let G_{0}(x) ∈ 𝒮_{0} be an invertible power series. If we choose a square root \sqrt{a_{0}} of a_{0} = G_{0}(0), then \sqrt{G_{0}(x)} is uniquely determined and belongs to 𝒮_{0} ⊗_{F} F (\sqrt{a_{0}}). Further if G_{0}(x) ∈ 𝒮_{0}(A_{c})_{2k}^{+}, then it is easy to verify that \sqrt{G_{0}(x)} ∈ 𝒮_{0}(A_{c})_{k}^{+}⊗_{F} F (\sqrt{a_{0}}). As an example, since \frac{1}{1 − 4x} ∈ 𝒮_{0}(A_{−4})_{2}^{+}, we have

\frac{1}{\sqrt{1 − 4x}} ∈ 𝒮_{0}(A_{−4})_{1}^{+},

whose n-th coefficient is \binom{2n}{n}. Let C = \{C_{n}\} be the Catalan numbers ([10, 6.2.1 Proposi-

tion]) whose generating function is known to be

G_{0}(C, x) = \frac{1 − \sqrt{1 − 4x}}{2x}.

We can show that G_{0}(C, x)^{2} ∈ 𝒮_{0}(A_{−4})_{1}^{+}. Since it is well known that G_{0}(C, x)^{2} =

\frac{G_{0}(C, x) − 1}{x}, we have \{C_{n+1}\}_{n≥0} ∈ 𝒮 (A_{−4})_{1}^{+}.

The generating function of Motzkin numbers is another example:

\frac{1 − x − \sqrt{1 − 2x − 3x^{2}}}{2x} ∈ 𝒮_{0}(A_{−2})_{0}^{−}.

Example 20. There are also examples of transcendental generating functions. Let H be the sequence of harmonic numbers ([7, Section 6.3]):

[0]H = 0, [n]H = 1 + \frac{1}{2} + ⋯ + \frac{1}{n}.

We have

G_{0}(H, x) = \frac{1}{1 − x} log (\frac{1}{1 − x}) ∈ 𝒮_{0}(A_{−1})_{2}^{−}.

More generally, Pfaff’s reflection law for the Gaussian hypergeometric function ([7, (5.101)])

\frac{1}{(−x + 1)^{α}} F ( {α, β \atop γ} | \frac{−x}{−x + 1}) = F ( {α, γ − β \atop γ} | x)

implies

F ( {α, β \atop 2β} | x) ∈ 𝒮_{0}(A_{−1})_{α}^{+}.

We next proceed to find examples of exponential generating functions of eigensequences using (11).

Example 21. We begin with examples of polynomials of exp(x). It is possible to determine all such polynomials giving eigensequences. First of all, a monomial generating function exp(cx) gives

⟨c⟩ ∈ 𝒮 (A_{−2c})_{1}^{+}

as we saw in Example 17.

We say that a polynomial P (t) ∈ F [t] is self-reciprocal if u^{deg P} P (t/u) ∈ F [t, u] is a symmetric polynomial: namely, setting d = deg P, it satisfies

u^{d}P (t/u) = t^{d}P (u/t).

Also a polynomial P (t) is called skew self-reciprocal if u^{deg P} P (t/u) ∈ F [t, u] is an alternating polynomial.

Let P (t) be a self-reciprocal polynomial. Then, by setting u = 1, t = exp(x) in the above formula, we obtain P (exp(x)) = exp(dx)P (exp(−x)). This means that G_{1}(x) = P (exp(x)) satisfies (11) with c = − deg P. Similarly using skew self-reciprocal polynomials, we are able to construct minus eigensequences.

The simplest examples of this construction are the binomials P (t) = t^{c} ± 1, which give

G_{1}(a^{±}, x) = exp(cx) ± 1 ∈ 𝒮_{1}(A_{−c})_{1}^{±}.

Example 22. We can directly verify using (2) that the Bernoulli numbers B = {B_{n}} satisfies

B ∈ 𝒮 (A_{1})_{1}^{+}.

Similarly, the Genocchi numbers G = {G_{n}} ([10, Exercise 5.8 d]) defined by

G_{1}(G, x) = \frac{2x}{exp(x) + 1}

belongs to 𝒮_{1}(A_{1})_{1}^{−}.

Example 23. Kummer’s first formula for confluent hypergeometric function (there is a typo in [2, (4.1.11)])

F ( {α \atop γ}| x) = exp(x) F ( {γ − α \atop γ}| − x)

leads to

F ( {α \atop 2α}| x) ∈ 𝒮_{1}(A_{−1})_{1}^{+}.

We define polynomial sequences associated to a sequence.

Definition 24. For any a ∈ 𝒮, we define the associated polynomial sequence P (a, t)_{k} of weight k by

P_{n}(a, t)_{k} = [n]P (a, t) = [n](a|_{[A_{t}]_{k}}) = (−1)^{n} ∑_{j=0}^{n} \binom{n + k − 1}{j}a_{n−j}t^{j} ∈ F [t].

Z.-W. Sun [13] and others adopted a slightly different definition. In our notation, their polynomial is (⟨−1⟩a) |_{[A_{t}A_{0}]_{1}}. Our definition seems to be more natural from our point of view.

The following proposition immediately follows from Proposition 14.

Proposition 25. If a ∈ 𝒮 (A_{c})_{k}^{s}, then P (a, t)_{k} ∈ 𝒮 (A_{2t−c})_{k}^{s}.

Example 26. A modified eigensequence ⟨−1⟩P (B, t)_{1} ∈ 𝒮 (A_{1−2t})_{1}^{+} of polynomials associated to the Bernoulli sequence B is that of the Bernoulli polynomials.

The Euler polynomials E_{n}(x) are defined by

\frac{2e^{tx}}{e^{x} + 1} = ∑_{n=0}^{∞} E_{n}(t)\frac{x^{n}}{n!} .

The Euler numbers E (see Example 6) are related to the polynomials by E_{n} (\frac{1}{2}) = \frac{E_{n}}{(−2)^{n}}.

It is easy to see that E′ = \{\frac{E_{n}}{(−2)^{n}} \} ∈ 𝒮 (A_{0})_{1}^{+}. The associated polynomials to E′ are given by

P_{n}(E′, t)_{1} = (−1)^{n}E_{n} (t + \frac{1}{2})

and it belongs to 𝒮 (A_{2t})_{1}^{+}.

The coefficients α_{2i} in (18) are closely related to associated polynomials. To show this relation, we first prove the following proposition, which generalizes well-known symmetry formulas for Bernoulli and Euler polynomials.

Proposition 27. If a ∈ 𝒮 (A_{c})_{k}^{s}, then we have

(−1)^{n}P_{n}(a, c − t)_{k} = sP_{n}(a, t)_{k}.

Proof. By Proposition 25, the sequence P (a, t)_{k} belongs to 𝒮 (A_{2t−c})_{k}^{s}. Therefore we have [n]P (a, t)_{k}|_{[A_{2t−c}]_{k}} = sP_{n}(a, t)_{k}. On the other hand, using the definition of the associated polynomials, we compute

[n]P (a, t)_{k}|_{[A_{2t−c}]_{k}} = [n] (a|_{[A_{t}]_{k}}) |_{[A_{2t−c}]_{k}} = [n] (a|_{[A_{t}A_{2t−c}]_{k}}) = [n] (a|_{[A_{c−t}A_{0}]_{k}}) = [n] (P (a, c − t))_{k} |_{[A_{0}]_{k}} = (−1)^{n}P_{n}(a, c − t)_{k}.

Here we used a similar method as in Proposition 14. This completes the proof.

Now we can show the following proposition.

Proposition 28. Let a ∈ 𝒮 (A_{c})_{k}^{±}. We have an explicit formula for (18) and (19):

G_{0}(a, x) = ∑_{i=0}^{∞} P_{i} (a, \frac{c}{2})_{k} x^{i}ℰ_{k+i}(x).

Proof. First note that the central value t = c/2 is a root of P_{n}(a, t) for odd n if a ∈ 𝒮 (A_{c})_{k}^{+} and for even n if a ∈ 𝒮 (A_{c})_{k}^{−}. This fact readily follows from Proposition 27. We calculate the coefficient of x^{m} in the right hand side of the equality using (4):

∑_{j=0}^{m} \binom{m + k − 1}{j}P_{m−j} (a, \frac{c}{2})_{k} (−\frac{c}{2})^{j} = [m] ((a|_{[A_{c/2}]_{k}})|_{[A_{c/2}]_{k}})) .

Since [A_{c/2}]_{k} is an involution, this is equal to [m]a. This proves the proposition.

Example 29. As an example, we compute the explicit expansion of the generating function G_{0}(F , x) ∈ 𝒮_{0}(A_{−1})_{1}^{−} of Fibonacci numbers. It is easy to compute

∑_{n≥0} P_{n} (F , −\frac{1}{2})_{1} x^{n} = G_{0}|_{[A_{−1/2}]_{1}}(F , x) = \frac{−x}{1 − \frac{5}{4}x^{2}} .

Therefore we get

G_{0}(F , x) = ∑_{i=0}^{∞} (\frac{5}{4})^{i} x^{2i+1}ℰ_{2i+2}(x).

Taking [x^{n}] of the both sides, we obtain a classical formula due to Catalan:

F_{n} = \frac{1}{2^{n−1}} ∑_{i=0}^{⌊ \frac{n−1}{2} ⌋} 5^{i} \binom{n}{n − 2i − 1}.

Example 30. There is another important family of polynomial eigensequences. They are orthogonal polynomials. The Jacobi polynomial P_{n}^{(α,β)}(t) is defined in terms of Gaussian hypergeometric function ([2, Definition 2.5.1]) and its exponential generating function is

\frac{2^{α+β}}{R(1 − x + R)^{α}(1 + x + R)^{β}}

(see [2, Theorem 6.4.2]) where R is defined by

R = \sqrt{1 − 2tx + x^{2}},

which belongs to 𝒮_{0}(A_{−2t})_{−1}^{+} by Examples 16 and 19. The product (1 − x + R)(1 + x − R) = 2(R+(1−xt)) is also in 𝒮_{0}(A_{−2t})_{−1}^{+}. Thus if α = β, then the exponential generating function belongs to 𝒮_{0}(A_{−2t})_{α+1}^{+}. We conclude

\{\frac{P_{n}^{(α,α)}(t)}{n!}\}_{n≥0} ∈ 𝒮 (A_{−2t})_{α+1}^{+}.

When α = β = 0, the polynomial P_{n}(t) = P_{n}^{(0,0)}(t) is called the Legendre polynomial. The following orthogonal polynomials also form polynomial eigensequences:

• The Gegenbauer polynomial C_{n}^{λ}(t) ([2, p.302]).

G_{0}(\{C_{n}^{λ}(t)\}, x) = \frac{1}{R^{2λ}} ∈ 𝒮_{0}(A_{−2t})_{2λ}^{+}.

The special case U_{n}(t) = C_{n}^{1}(t) is called Chebyshev polynomial of second kind.

• The Chebyshev polynomial of first kind T_{n}(t) ([2, Remark 2.5.3]). The ordinary generating function is

\frac{1 − tx}{R^{2}} ∈ 𝒮_{0}(A_{−2t})_{1}^{+}.

• The Hermite polynomial H_{n}(t) ([2, Section 6.1]). The exponential generating function is exp(2tx − t^{2}) ∈ 𝒮_{1}(A_{−4t})_{1}^{+}.

4 Differential operations on eigensequences

The main result in this section is the following analogue of the Rankin-Cohen differential operator. Since the sign differs from the original formula (see [14, Section 1]), we state and

prove the following theorem for a general lower triangular matrix A = [\begin{matrix}a & 0 \\ c & d \end{matrix}].

Theorem 31. Let n be a non-negative integer and k and ℓ integers such that the following conditions are not satisfied:

−n < k ≤ 0 or − n < ℓ ≤ 0.  (20)

For f ∈ 𝒮_{0}(A)_{k}^{s_{1}} and g ∈ 𝒮_{0}(A)_{ℓ}^{s_{2}}, we define the n-th Rankin-Cohen bracket of f and g by the formula

[f, g]_{n}(x) = ∑_{r+s=n}(−1)^{r}\binom{n + k − 1}{s}\binom{n + ℓ − 1}{r}f^{(r)}(x)g^{(s)}(x).

Then we have

[f, g]_{n} ∈ 𝒮_{0}(A)k+ℓ+2n^{\frac{s_{1}s_{2}}{(ad)^{n}}}.

Proof. Our formulation and proof are based on Zagier’s article [14]. Therefore we only give a sketch of the proof. However we have to take care of negative k and ℓ. We first associate a formal power series f̃(x, t) to f(x)

f̃(x, t) = ∑_{i=0}^{n} \frac{f^{(i)}(x)}{(i + k − 1)\underline{i}} \frac{t^{i}}{i!}.

The formal power series g̃(x, t) for g(x) is similarly defined. This definition makes sense if (20) are not satisfied.

Then we can prove

[t^{n}]f̃(x, −t)g̃(x, t) = \frac{[f, g]_{n}(x)}{(n + k − 1)\underline{n}(n + ℓ − 1)\underline{n}}

On the other hand, we can show the following formulas for the higher derivatives by induction:

\frac{(ad)^{n}f^{(n)} (\frac{ax}{cx+d})}{n!(n + k − 1)\underline{n}} = s_{1} ∑_{m=0}^{n} \frac{c^{n−m}(cx + d)^{k+n+m}}{(n − m)!} \frac{f^{(m)}(x)}{m!(m + k − 1)\underline{m}} .

It follows from these formulas that f̃ satisfies the transformation law

f̃(\frac{ax}{cx + d}, \frac{(ad)t}{(cx + d)^{2}} ) ≡ s_{1}(cx + d)^{k} exp (\frac{ct}{cx + d}) f̃(x, t) (mod t^{n+1}).

A similar formula holds also for g̃:

g̃ (\frac{ax}{cx + d}, \frac{(ad)t}{(cx + d)^{2}} ) ≡ s_{2}(cx + d)^{ℓ} exp (\frac{ct}{cx + d}) g̃(x, t) (mod t^{n+1}).

Multiplying the both sides, we have

f̃(\frac{ax}{cx + d}, −\frac{(ad)t}{(cx + d)^{2}} ) g̃ (\frac{ax}{cx + d}, \frac{(ad)t}{(cx + d)^{2}} ) ≡ s_{1}s_{2}(cx+d)^{k+ℓ}f̃(x, −t)g̃(x, t) (mod t^{n+1}).

Comparing the coefficients of t^{n}, we obtain

[f, g]_{n} (\frac{ax}{cx + d}) = \frac{s_{1}s_{2}}{(ad)^{n}} (cx + d)^{k+ℓ+2n}[f, g]_{n}(x).

This means that [f, g]_{n} ∈ 𝒮_{0}(A)k+ℓ+2n^{\frac{s_{1}s_{2}}{(ad)^{n}}}. This completes the proof of the theorem.

As in the paper [14], the following identities involving the brackets hold for f ∈ 𝒮_{0}(A)_{k}^{s_{1}}, g ∈ 𝒮_{0}(A)_{ℓ}^{s_{2}} and h ∈ 𝒮_{0}(A)_{m}^{s_{3}}:

[f, g]_{n} = (−1)^{n}[g, f]_{n},
[[f, g]_{1}, h]_{1} + [[g, h]_{1}, f]_{1} + [[h, f]_{1}, g]_{1} = 0 (the Jacobi identity),
[[f, g]_{0}, h]_{1} + [[g, h]_{0}, f]_{1} + [[h, f]_{0}, g]_{1} = 0,
m[[f, g]_{1}, h]_{0} + k[[g, h]_{1}, f]_{0} + ℓ[[h, f]_{1}, g]_{0} = 0,
[[f, g]_{0}, h]_{1} = [[g, h]_{1}, f]_{0} + [[h, f]_{1}, g]_{0},
(k + m + ℓ)[[f, g]_{1}, h]_{0} = k[[h, f]_{0}, g]_{1} − ℓ[[g, h]_{0}, f]_{1}.

Here we write down the brackets of low degree in the case of A = A_{c}:

[f, g]_{0} = fg ∈ 𝒮_{0}(A_{c})_{k+ℓ}^{s_{1}s_{2}},
[f, g]_{1} = kfg′ − ℓf′g ∈ 𝒮_{0}(A_{c})_{k+ℓ+2}^{−s_{1}s_{2}},
[f, g]_{2} = \frac{ℓ(ℓ + 1)}{2}f′′g − (k + 1)(ℓ + 1)f′g′ + \frac{k(k + 1)}{2}fg′′ ∈ 𝒮_{0}(A_{c})_{k+ℓ+4}^{s_{1}s_{2}}.

Example 32. Let a = {a_{i}} ∈ 𝒮 (A_{c})_{k}^{s} (k > 0). Since x^{ℓ} ∈ 𝒮 (A_{c})_{−ℓ}^{(−1)^{ℓ}} (see (17)), we have

[G_{0}(a, x), x^{ℓ}]_{n} = ∑_{s=0}^{n}(−1)^{n−s}\binom{n + k − 1}{s}\binom{n − ℓ − 1}{n − s} (∑_{i=0}^{∞}(i + n − s)\underline{n−s}a_{i+n−s}x^{i}) ℓ\underline{s} x^{ℓ−s}

= ∑_{i=0}^{∞} (∑_{s=0}^{n}(−1)^{n−s}\binom{n + k − 1}{s}\binom{n − ℓ − 1}{n − s}ℓ\underline{s}(i + n − ℓ)\underline{n−s}) a_{i+n−ℓ}x^{i}

= ∑_{i=0}^{∞} (∑_{s=0}^{n}(−1)^{n−s}\binom{n + k − 1}{s}(−1)^{n−s}\binom{ℓ − s}{n − s}s!\binom{ℓ}{s}(n − s)!\binom{i − ℓ + n}{n − s}) a_{i+n−ℓ}x^{i}

= n! \binom{ℓ}{n} ∑_{i=0}^{∞} (∑_{s=0}^{n} \binom{n + k − 1}{s}\binom{i − ℓ + n}{n − s}a_{i+n−ℓ}) x^{i}

= n! \binom{ℓ}{n} ∑_{i=0}^{∞} \binom{2n + k + i − ℓ − 1}{n}a_{i+n−ℓ}x^{i} ∈ 𝒮_{0}(A_{c})_{k−ℓ+2n}^{s(−1)^{n+ℓ}}.

for n ≤ ℓ. Here the last equality follows from Vandermonde’s convolution. An interesting case is ℓ = 2n. Then [∗, x^{2n}]_{n} preserves the weight and we have

[G_{0}(a, x), x^{2n}]_{n} = (2n)\underline{n} ∑_{i=n}^{∞} \binom{i + k − 1}{n} a_{i−n}x^{i} ∈ 𝒮_{0}(A_{c})_{k}^{(−1)^{n}s}.

We conclude that

\{\binom{i + k − 1}{n} a_{i−n}\}_{i≥0} ∈ 𝒮 (A_{c})_{k}^{(−1)^{n}s}

and

\{(i + k − 1)\underline{n} a_{i−n}\}_{i≥0} ∈ 𝒮 (A_{c})_{k}^{(−1)^{n}s}.

There is also an analogue of Theorem 31 for exponential generating functions.

Theorem 33. Let n be a non-negative integer. If f ∈ 𝒮_{1}(A_{c_{1}})_{1}^{s_{1}} and g ∈ 𝒮_{1}(A_{c_{2}})_{1}^{s_{2}}, then

\{f, g\}_{n} = ∑_{j=0}^{n} \binom{n}{j}c_{2}^{j}(−c_{1})^{n−j}f^{(j)}(x)g^{(n−j)}(x) ∈ 𝒮_{1}(A_{c_{1}+c_{2}})_{1}^{(−1)^{n}s_{1}s_{2}}.

Proof. Using Leibniz’s rule, we calculate

exp(−(c_{1} + c_{2})x) ∑_{j=0}^{n} \binom{n}{j}c_{2}^{j}(−c_{1})^{n−j}f^{(j)}(−x) ⊗ g^{(n−j)}(−x)

= exp(−(c_{1} + c_{2})x) ∑_{j=0}^{n} \binom{n}{j}c_{2}^{j}(−c_{1})^{n−j}

× (s_{1} exp(c_{1}x) (−c_{1} − \frac{d}{dx})^{j} f(x)) ⊗ (s_{2} exp(c_{2}x) (−c_{2} − \frac{d}{dx})^{n−j} g(x))

= s_{1}s_{2} ∑_{j=0}^{n} \binom{n}{j}c_{2}^{j}(−c_{1})^{n−j} ((−c_{1} − \frac{d}{dx})^{j} f(x) ⊗ (−c_{2} − \frac{d}{dx})^{n−j} g(x))

= s_{1}s_{2} (c_{2} (−c_{1} − \frac{d}{dx}) + (−c_{1}) (−c_{2} − \frac{d}{dx}))^{n} f(x) ⊗ g(x)

= (−1)^{n}s_{1}s_{2} (c_{2} \frac{d}{dx} − c_{1} \frac{d}{dx})^{n} f(x) ⊗ g(x)

= (−1)^{n}s_{1}s_{2} ∑_{j=0}^{n} \binom{n}{j}c_{2}^{j}(−c_{1})^{n−j}f^{(j)}(x)g^{(n−j)}(x).

This implies that

exp(−(c_{1} + c_{2})x){f, g}_{n}(−x) = (−1)^{n}s_{1}s_{2}{f, g}_{n}(x).

The result follows from (11).

If f(x) = G_{1}(a, x) and g(x) = G_{1}(b, x), then we compute

[\frac{x^{m}}{m!} ] \{G_{1}(a, x), G_{1}(b, x)\}_{n} = ∑_{j=0}^{n} \binom{n}{j} c_{2}^{j}(−c_{1})^{n−j} ∑_{i=0}^{m} \binom{m}{i} a_{i+j}b_{m+n−i−j}.  (21)

In particular, if c_{2} = −c_{1}, then

[\frac{x^{m}}{m!} ] \{G_{1}(a, x), G_{1}(b, x)\}_{n} = (−c_{1})^{n} [\frac{x^{m+n}}{(m + n)!}] \{G_{1}(a, x), G_{1}(b, x)\}_{0}.

We again write down the brackets of low degree:

{f, g}_{0} = fg ∈ 𝒮_{1}(A_{c_{1}+c_{2}})_{1}^{s_{1}s_{2}},
{f, g}_{1} = −c_{1}fg′ + c_{2}f′g ∈ 𝒮_{1}(A_{c_{1}+c_{2}})_{1}^{−s_{1}s_{2}},
{f, g}_{2} = c_{1}^{2}fg′′ − 2c_{1}c_{2}f′g′ + c_{2}^{2}f′′g ∈ 𝒮_{1}(A_{c_{1}+c_{2}})_{1}^{s_{1}s_{2}}.

These brackets also satisfy the analogous properties of the Rankin-Cohen brackets. We omit to write down these formulas.

We conclude this section with a few remarks.

We can apply Theorem 33 only to eigensequences of weight 1. Thus it is worth considering a method to transform a general eigensequence to a weight-1 sequence.

The simplest method is shifting. Namely by shifting an eigensequence of positive weight to the right, it eventually becomes a sequence of weight 1. The right shifts correspond to iterated formal integration of the exponential generating function (cf. Proposition 2).

There is another interesting method. Let a ∈ 𝒮 (A_{c})_{k}^{±} (c ≠ 0). We choose integer parameters α and β and define a new sequence b by

G_{1}(b, x) = exp(βx)G_{0} (a, \frac{1}{c}(exp(αx) − 1)) .  (22)

An easy calculation shows that

G_{1}(b, x) = ± exp((2β − kα)x)G_{1}(b, −x)  (23)

holds. By (11) we conclude b ∈ 𝒮 (A_{kα−2β})_{1}^{±}.

The transformation (22) is closely related to a generalization of the Akiyama-Tanigawa algorithm (see [1] and [9]). We explain this connection briefly. Starting from the initial sequence a = {a_{0,m}}_{m≥0}, we define new sequences {a_{n,m}}_{m≥0} for n = 1, 2, … by

a_{n,m} = (αm + β)a_{n−1,m} + (γm + δ)a_{n−1,m+1}  (24)

with given parameters α, β , γ and δ. Then we collect the 0-th terms and form a sequence b = {a_{n,0}}_{n≥0}. If δ = γ, then by a similar method as [4], we can prove

a_{n,0} = ∑_{j=0}^{n} \binom{n}{j}β^{n−j} (∑_{m=0}^{n} \left\{{n \atop m}\right\}α^{n−m}γ^{m}m! a_{0,m}) ,

where \left\{{n \atop m}\right\} is the Stirling number of the second kind and

G_{1}(b, x) = exp(βx)G_{0} (a, \frac{γ}{α}(exp(αx) − 1)) .

Chen [4] proved these formulas for the special cases where (α, β, γ) = (1, 1, −1) and (1, 0, −1). We have chosen γ = α/c in (24). Incidentally, if α = γ = 0, then the sequences {a_{n,m}} form a matrix called generalized

Seidel matrix. In this case, assuming δ ≠ 0, we have

G_{1}(a, x) = exp (−\frac{βx}{δ}) G_{1} (b, \frac{x}{δ})

(see [5, Theorem 2.1]). Hence this transformation can be written by our action (but it is not a convolution). Moreover, if a ∈ 𝒮 (A_{c})_{1}^{s}, then we can prove b ∈ 𝒮 (A_{cδ−2β})_{1}^{s}.

5 Involutive action on endomorphisms

In this section, we study the action of the involutions on the endomorphism algebra End(𝒮) of the vector space 𝒮.

Definition 34. Let A ∈ GL_{2}(F) be a lower triangular regular matrix of order 2. We define a left action of A of weight k on End(𝒮) by

([A]_{k}f)(a) = (f(a|_{[A]_{k}}))|_{[A]_{k}}  (25)

for f ∈ End(𝒮) and a ∈ 𝒮.

Since A acts as an involution on End(𝒮), we can decompose End(𝒮) as an F [A]_{k}-module:

End(𝒮) = End(𝒮)^{+} ⊕ End(𝒮)^{−},

where the eigenspaces are given by

End(𝒮)^{+} = {f ∈ End(𝒮) | [A]_{k}f = f}, End(𝒮)^{−} = {f ∈ End(𝒮) | [A]_{k}f = −f}.

It is easy to see that End(𝒮)^{+} coincides with the submodule End_{F [A]_{k}}(𝒮) of F [A]_{k}-endomorphisms. From this fact, we can show the following proposition easily.

Proposition 35. If a^{+} ∈ 𝒮 (A)_{k}^{+}, a^{−} ∈ 𝒮 (A)_{k}^{−}, f^{+} ∈ End(𝒮)^{+}andf^{−} ∈ End(𝒮)^{−}, then we have

f^{+}(a^{+}) ∈ 𝒮 (A)_{k}^{+}, f^{+}(a^{−}) ∈ 𝒮 (A)_{k}^{−}, f^{−}(a^{+}) ∈ 𝒮 (A)_{k}^{−}, f^{−}(a^{−}) ∈ 𝒮 (A)_{k}^{+}.

This proposition gives another method to generate new eigensequences.

We shall compute the action of [A_{c}]_{k} on f ∈ End(𝒮) explicitly for several elementary endomorphisms f.

In the following, until the end of this section, we restrict ourselves to the case where the weight k = 1, for simplicity.

Proposition 36. Let ∂^{m} be the differentiation on exponential generating function:

∂^{m} : G_{1}(a, x) ↦ \frac{d^{m}}{dx^{m}} G_{1}(a, x).

Then we have

[A_{c}]_{1}∂^{m}(G_{1}(a, x)) = (−c − ∂)^{m} G_{1}(a, x).  (26)

Proof. Applying Leibniz’s rule to (8), we have

∂^{m}G_{1}(a|_{[A_{c}]_{1}}, x) = ∑_{i=0}^{m} \binom{m}{i}(−c)^{m−i} exp(−cx) (−1)^{i}∂^{i} G_{1}(a, −x)

= (−1)^{m} ∑_{i=0}^{m} \binom{m}{i} c^{m−i} exp(−cx) ∂^{i} G_{1}(a, −x)

Therefore we get

[A_{c}]_{1}∂^{m}(G_{1}(a, x)) = exp(−cx)(−1)^{m} ∑_{i=0}^{m} \binom{m}{i}c^{m−i} exp(cx) ∂^{i} G_{1}(a, x)

= (−1)^{m} ∑_{i=0}^{m} \binom{m}{i}c^{m−i}∂^{i} G_{1}(a, x)

= (−c − ∂)^{m}G_{1}(a, x).

We rewrite this result in terms of sequence. Proposition 37. Let a ∈ 𝒮 and f = L^{m} the map shifting m (m ≥ 1) terms to the left:

[n](L^{m}(a)) = a_{m+n}, [0](L^{m}(a)) = a_{m}.

Then we have

[n](([A_{c}]_{1}L^{m})(a)) = (−1)^{m} ∑_{i=0}^{m} \binom{m}{i}c^{m−i}a_{n+i} = [m](L^{n}(a)|_{[A_{c}]_{1}}),
[0](([A_{c}]_{1}L^{m})(a)) = (−1)^{m} ∑_{i=0}^{m} \binom{m}{i}c^{m−i}a_{i} = [m](a|_{[A_{c}]_{1}}).

Proof. We obviously have ∂^{m}G_{1}(a, x) = G_{1}(L^{m}a, x). Hence the result follows by taking the n-th terms of (26). Proposition 38. Let f = Δ be the difference operator defined by a_{n} ↦ a_{n+1} − a_{n}. Then we have

[n](([A_{c}]_{1}Δ^{m})(a)) = (−1)^{m} ∑_{j=0}^{m} \binom{m}{j}(1 + c)^{m−j}a_{n+j} = [m](L^{n}(a)|_{[A_{1+c}]_{1}}),
[0](([A_{c}]_{1}Δ^{m})(a)) = (−1)^{m} ∑_{j=0}^{m} \binom{m}{j}(1 + c)^{m−j}a_{j} = [m](a|_{[A_{1+c}]_{1}}).

Proof. It is easy to show that

[n](Δ^{m}(a)) = ∑_{i=0}^{m} \binom{m}{i}(−1)^{m+i}a_{n+i}.

Hence we have

Δ^{m}(G_{0}(a|_{[A_{c}]_{1}}, x)) = ∑_{i=0}^{m} \binom{m}{i}(−1)^{m+i}L^{i}G_{0}(a|_{[A_{c}]_{1}}, x)).

From the definition (25) it follows that

([A_{c}]_{1}Δ^{m})(G_{0}(a, x)) = ∑_{i=0}^{m} \binom{m}{i}(−1)^{m+i}([A_{c}]_{1}L^{i})G_{0}(a, x).

Therefore, by Proposition 37, the n-th term is given by

[n](([A_{c}]_{1}Δ^{m})(a)) = (−1)^{m} ∑_{i=0}^{m} ∑_{t=0}^{i} \binom{m}{i}\binom{i}{t}c^{t}a_{n+i−t}

= (−1)^{m} ∑_{j=0}^{m} ∑_{i=j}^{m} \binom{m}{i}\binom{i}{j}c^{i−j}a_{n+j}.

The binomial identity \binom{m}{i}\binom{i}{j} = \binom{m}{j}\binom{m − j}{m − i} yields

[n](([A_{c}]_{1}Δ^{m})(a)) = (−1)^{m} ∑_{j=0}^{m} \binom{m}{j} ∑_{i=0}^{m−j} \binom{m − j}{i}c^{i}a_{n+j} = (−1)^{m} ∑_{j=0}^{m} \binom{m}{j}(1 + c)^{m−j}a_{n+j}

as desired.

Another differential operators producing new eigensequences are given in the following corollary, whose proof readily follows from Propositions 35 and 36. Corollary 39. Let G(x) ∈ 𝒮_{1}(A_{c})_{1}^{t}. For any polynomial p(∂) ∈ F [∂], we define

p(∂)G(x) = p (\frac{d}{dx}) G(x).

If p(∂) ∈ F [∂] satisfies

p(∂) = s · p(−c − ∂)  (27)

with some s ∈ {±1}, then we have

p(∂)G(x) ∈ 𝒮 (A_{c})_{1}^{st}.

The following lemma tells us how to find polynomials satisfying (27).

Lemma 40. Let u(∂) be an even (resp. odd) polynomial in ∂. Then p(∂) = u (∂ + \frac{c}{2}) satisfies (27) with s = 1 (resp. s = −1). Conversely every polynomial satisfying (27) is obtained in this way.

Proof. The first half follows from an easy calculation. We shall prove the latter half. Notice that the map ∂ ↦ −c − ∂ is an involution on F [∂]. Therefore any polynomial p satisfying (27) is obtained by p(∂) = h(∂) + s h(−c − ∂) for some h(∂) ∈ F [∂]. Then we have

p (∂ − \frac{c}{2}) = h (∂ − \frac{c}{2}) + s h (−∂ − \frac{c}{2}) .

This shows that p (∂ − \frac{c}{2}) is an even (resp. odd) polynomial if s = 1 (resp. s = −1).

Example 41. We give some examples of polynomials satisfying (27). First we use u(∂) in Lemma 40 to obtain

\frac{c}{2} + ∂ (s = −1),  (28)
c∂ + ∂^{2} (s = 1).  (29)

Although we have (∂ + \frac{c}{2})^{2} = ∂^{2} + c∂ + \frac{c^{2}}{4} , we can drop the constant term since 1 ∈ F [∂] defines an operator with s = 1. These two operators are found in [12, Corollary 3.1] as operators related to [A_{−1}]_{1}.

We have another type of operators like

∂^{n} ± (−c − ∂)^{n} ,  (30)
∂^{n}(−c − ∂)^{m} ± (−c − ∂)^{n}∂^{m},  (31)
(∂^{2} − c^{2})^{n} ± (∂^{2} + 2c∂)^{n}  (32)

with s = ±1 and m, n ∈ ℤ_{>0}.

These operators will be used to produce identities of eigensequences in the next section.

6 Identities for eigensequences

In this section, we shall deduce some identities involving eigensequences as an application of the theory we have developed so far in the preceding sections.

First we show that we can construct eigensequences from arbitrary sequences easily by applying ‘twisted’ endomorphisms.

Proposition 42. Let f ∈ End(𝒮) and a ∈ 𝒮. Let A ∈ GL_{2}(F) be a lower triangular matrix of order 2. Then we have

1. f(a) + ([A]_{k}f)(a|_{[A]_{k}}) ∈ 𝒮 (A)_{k}^{+};

2. f(a) − ([A]_{k}f)(a|_{[A]_{k}}) ∈ 𝒮 (A)_{k}^{−}; 3. ([A]_{k}f)(a) + f(a|_{[A]_{k}}) ∈ 𝒮 (A)_{k}^{+};

4. ([A]_{k}f)(a) − f(a|_{[A]_{k}}) ∈ 𝒮 (A)_{k}^{−}.

Proof. The assertions 3 and 4 follow from 1 and 2 respectively taking [A]_{k}f as f. Therefore we have only to prove the first two assertions. Recall that we have projections π(A)_{k}^{+} (resp. π(A)_{k}^{−}) from 𝒮 to 𝒮 (A)_{k}^{+} (resp. 𝒮 (A)_{k}^{+}) (see the remark after Proposition 14). By Definition 34, we have

f(a) + ([A]_{k}f)(a|_{[A]_{k}}) = f(a) + f(a)|_{[A]_{k}} = 2π(A)_{k}^{+}(f(a)) ∈ 𝒮 (A)_{k}^{+}.

Similarly, we get

f(a) − ([A]_{k}f)(a|_{[A]_{k}}) = 2π(A)_{k}^{−}(f(a)) ∈ 𝒮 (A)_{k}^{−}.

This completes the proof.

We have the following corollary.

Corollary 43. Let f, g ∈ End(𝒮) and a ∈ 𝒮. Then we have

1. ([A]_{k}f ∘ g)(a) + (f ∘ [A]_{k}g)(a|_{[A]_{k}}) ∈ 𝒮 (A)_{k}^{+};

2. ([A]_{k}f ∘ g)(a) − (f ∘ [A]_{k}g)(a|_{[A]_{k}}) ∈ 𝒮 (A)_{k}^{−}.

Proof. We first compute the action of [A_{c}]_{k} in f ∘ g:

([A_{c}]_{k}(f ∘ g))(a) = f(g(a|_{[A_{c}]_{k}}))|_{[A_{c}]_{k}} = ([A_{c}]_{k}f)(g(a|_{[A_{c}]_{k}})|_{[A_{c}]_{k}}) = ([A_{c}]_{k}f) ∘ ([A_{c}]_{k}g)(a).

Consequently we have [A_{c}]_{k}([A_{c}]_{k}f ∘g) = f ∘[A_{c}]_{k}g. Hence the results follow from Proposition 42.

Among various properties of eigensequences, one of the easiest is (14):

[0]a = 0 if a ∈ 𝒮^{−}.

Thus by Proposition 42 2, 4 and Corollary 43 the following equalities hold for any sequence a ∈ 𝒮 and any f, g ∈ End(𝒮):

[0]f(a) = [0]([A_{c}]_{k}f)(a|_{[A_{c}]_{k}}),  (33)
[0]([A_{c}]_{k}f)(a) = [0]f(a|_{[A_{c}]_{k}}),  (34)
[0]([A_{c}]_{k}f ∘ g)(a) = [0](f ∘ [A_{c}]_{k}g)(a|_{[A_{c}]_{k}}).  (35)

For every f and g, we obtain an identity involving a. We give several explicit examples of such formulas in the following proposition. We restrict ourselves again to the case k = 1 for simplicity.

Proposition 44. Let a = {a_{n}} be any sequence and a_{n}^{∗} = [n](a|_{[A_{c}]_{1}}). Then the following identities hold.

1. ∑_{i=0}^{m} \binom{m}{i}(−1)^{i}a_{i} = ∑_{i=0}^{m} \binom{m}{i}(1 + c)^{m−i}a_{i}^{∗}.

2. (−1)^{n} ∑_{i=0}^{n} \binom{n}{i}c^{n−i}a_{m+i} = (−1)^{m} ∑_{i=0}^{m} \binom{m}{i}c^{m−i}a_{n+i}^{∗}.

3. ∑_{i=0}^{m} \binom{m}{i}(2c)^{m−i}a_{m+i} = ∑_{i=0}^{m} \binom{m}{i}(−c^{2})^{m−i}a_{2i}^{∗}.

Note that the roles of a_{n} and a_{n}^{∗} are interchangeable.

Proof. Let a^{∗} = a|_{[A_{c}]_{1}}. The following identities are used to obtain the first two formulas:

1. [0](Δ^{m}(a)) = [0]([A_{c}]_{1}Δ^{m}(a^{∗})) (we take f = Δ^{m} in (33));

2. [0](L^{m} ∘ [A_{c}]_{1}L^{n}(a)) = [0]([A_{c}]_{1}L^{m} ∘ L^{n}(a^{∗})) (we take f = L^{m} and g = L^{n} in (35)).

Here we have already computed the explicit formulas for [A_{c}]_{1}Δ^{m} and [A_{c}]_{1}L^{m} in Propositions 37 and 38. The identity 3 is obtained by taking f = (L^{2} − c^{2})^{m} and using (34) since we have [n](L^{2} − c^{2})(a) = a_{n+2} − c^{2}a_{n} and [n]([A_{c}]_{1}L^{2} − c^{2})(a) = 2ca_{n+1} + a_{n+2}.

Note that 2 is a generalization of the formula due to Chen [5, Theorem 2.1] and Gessel [6, Theorem 7.4]. Kaneko’s recursion formula (2) is an easy consequence of 2 in the preceding proposition as is noted by Gessel [6, Lemma 7.2]. The third formula in the proposition seems to be not known before.

These identities will be simple and particularly interesting if a|_{[A_{c}]_{k}} is simple. This is the case if a is an eigensequence in 𝒮 (A_{c})_{k}^{±}.

In addition to eigensequences, there are some interesting pairs of a and a|_{[A_{c}]_{k}}:

1. a = {n!} and derangement numbers D(n) with [A_{−1}]_{1} ([6, Section 7]);

2. The special values ξ_{k}(−n) of negative integers of Arakawa-Kaneko zeta function and the poly-Bernoulli number B_{n}^{(k)} with [A_{−1}]_{1} ([3, Theorem 6]).

As we have seen in Proposition 5 and Lemma 15, the terms in eigensequences satisfy simple relations. These relations can be used to obtain identities for eigensequences.

Proposition 45. Assume that c ≠ 0. Let a ∈ 𝒮 (A_{c})_{1}^{s_{1}} and b ∈ 𝒮 (A_{−c})_{1}^{s_{2}}. Then we have

[n](a ∗_{1} b) = ∑_{i=0}^{n} \binom{n}{i} a_{i}b_{n−i} = 0

either s_{1}s_{2} = 1 and n is odd or s_{1}s_{2} = −1 and i is even.

Proof. By Proposition 11 we have a ∗_{1} b ∈ 𝒮 (A_{0})_{1}^{s_{1}s_{2}}. By Proposition 5, the exponential generating function of a∗_{1}b is an even (resp. odd) power series if s_{1}s_{2} = 1( resp. s_{1}s_{2} = −1). The proposition is now clear from this.

The identities

∑_{i=0}^{n} \binom{n}{i}B_{i}L_{n−i} = 0, if n is odd;
∑_{i=0}^{n} \binom{n}{i}B_{i}F_{n−i} = 0, if n is even

involving the Lucas, Bernoulli and Fibonacci numbers proved in [12, (4.1) and (4.2)] follow readily from Proposition 45

Along with various differential operators, we can obtain more identities.

Corollary 46. If a ∈ 𝒮 (A_{c})_{1}^{+} (resp. 𝒮 (A_{c})_{1}^{−}), then we have

c ∑_{j=0}^{n} \binom{n}{j}(−1)^{n−j}a_{j}a_{n−j} + 2 ∑_{j=0}^{n} \binom{n}{j}(−1)^{n−j}a_{j}a_{n−j+1} = 0

for even n (resp. odd n).

Proof. Let a ∈ 𝒮 (A_{c})_{1}^{+}. Then it follows from (28) that

(2 + c\frac{d}{dx}) G_{1} (a, x) ∈ 𝒮_{1}(A_{c})_{1}^{−}

and by Proposition 13 we have

exp(−x) · (2 + c\frac{d}{dx}) G_{1} (a, x) ∈ 𝒮_{1}(A_{−c})_{1}^{−}.

Therefore we obtain

\{G_{1}(a, x), exp(−x) · (\frac{c}{2} + \frac{d}{dx}) G_{1}(a, x)\}_{0} ∈ 𝒮_{1}(A_{0})_{1}^{−},

which is an odd power series by Proposition 5. The left hand side of the identity in the statement is the n-th term of this sequence (see (21)). This proves the assertion. The proof for a ∈ 𝒮 (A_{c})_{1}^{−} is similar.

Proposition 47. Let a ∈ 𝒮 (A_{c})_{1}^{s}. Choose a positive integer j such that (−1)^{j}s = −1. Then we have

∑_{i=0}^{n} \binom{n}{i} c^{n−i}(n + i)\underline{j} a_{n−j+i} = 0.  (36)

Proof. By Example 32, we have \{i\underline{j} a_{i−j}\}_{i≥0} ∈ 𝒮 (A_{c})_{1}^{−}. By applying c∂ +∂^{2} in (29) n times, the m-th term of the new sequence is

∑_{i=0}^{n} \binom{n}{i} c^{n−i}(n + i + m)\underline{j} a_{n+i+m−j}.

Since this sequence belongs to 𝒮 (A_{c})_{1}^{−}, the 0-th term is 0 by Lemma 15:

∑_{i=0}^{n} \binom{n}{i} c^{n−i}(n + i)\underline{j} a_{n+i−j} = 0.

The following corollary readily follows from (36) by setting j = 1.

Corollary 48 (Generalized Kaneko’s identity). If a ∈ 𝒮 (A_{c})_{1}^{+}, then

∑_{i=0}^{n} \binom{n}{i} c^{n−i}(n + i)a_{n+i−1} = 0.

By noting that the Bernoulli numbers B ∈ 𝒮 (A_{1})_{1}^{+} (see Example 22), the original formula (4) follows by changing n by n + 1.

As we have seen, many identities for eigensequences can be obtained from our study on the involutions instead of long and complicate computations of iterated sums.

We hope that our method using the involutions and the differential operators shed new light on the study of sequences.

References

[1] S. Akiyama and Y. Tanigawa, Multiple zeta values at non-positive integers, Ramanujan J. 5 (2001), 327–351.

[2] G. E. Andrews, R. Askey, and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999.

[3] T. Arakawa and M. Kaneko, Multiple zeta values, poly-Bernoulli numbers, and related zeta functions, Nagoya Math. J. 153 (1999), 189–209.

[4] K.-W. Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Seq. 4 (2001), Article 01.1.6.

[5] K.-W. Chen, Identities from the binomial transform, J. Number Theory 124 (2007), 142–150.

[6] I. M. Gessel, Applications of the classical umbral calculus, Algebra Universalis 49 (2003), 397–434.

[7] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics, second ed., Addison-Wesley Publishing Company, Reading, MA, 1994.

[8] M. Kaneko, A recurrence formula for the Bernoulli numbers, Proc. Japan Acad. Ser. A Math. Sci. 71 (1995), 192–193.

[9] M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Seq. 3 (2000), Article 00.2.9.

[10] R. P. Stanley, Enumerative Combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999.

[11] W. Stein, Modular Forms, a Computational Approach, Graduate Studies in Mathematics, vol. 79, American Mathematical Society, Providence, RI, 2007.

[12] Z.-H. Sun, Invariant sequences under binomial transformation, Fibonacci Quart. 39 (2001), 324–333.

[13] Z.-W. Sun, Combinatorial identities in dual sequences, European J. Combin. 24 (2003), 709–718.

[14] D. Zagier, Modular forms and differential operators, Proc. Indian Acad. Sci. Math. Sci. 104 (1994), 57–75.

[15] The On-Line Encyclopedia of Integer Sequences, published electronically at

    http://oeis.org, 2010.

2010 Mathematics Subject Classification: Primary 11B75; Secondary 11B37, 11B68, 05A19, 05A15. Keywords: generating function, differential operator, linear recurrence.

Received November 18 2012; revised version received January 16 2013. Published in Journal of Integer Sequences, January 17 2013.

Return to Journal of Integer Sequences home page.