Integer Sequences Connected to the Laplace Continued Fraction and Ramanujan’s Identity

Alexander Kreinin Risk Analytics IBM 185 Spadina Avenue Toronto, ON M5T 2C6 Canada

               alex.kreinin@ca.ibm.com

Abstract

We consider integer sequences connected to the famous Laplace continued fraction for the function R(t) = ∫_{t}^{∞} φ(x)dx/φ(t), where φ(t) = e^{−t^{2}/2}/\sqrt{2π} is the standard normal density. We compute the generating functions for these sequences and study their relation to the Hermite and Bessel polynomials. Using the master equation for the generating functions, we find a new proof of the Ramanujan identity.

1 Introduction

We consider two infinite matrices, 𝔭 = ∥p_{k,m}∥_{k,m≥0} and 𝔮 = ∥q_{k,m}∥_{k,m≥0}, defined as follows. If m > k or k − m ≡ 1 (mod 2), then

p_{k,m} = q_{k,m} = 0, k, m = 0, 1, ….

If k = m + 2n and n ≥ 0 then

p_{k,m} = \frac{k!}{m! 2^{n} n!},  (1)

and

q_{k,m} = \frac{(\frac{k+m}{2})!}{m!}2^{−n} ∑_{j=0}^{n} \binom{k + 1}{j}.  (2)

The matrices 𝔭 and 𝔮 are connected to the Laplace continued fraction [11]

L(t) := \frac{1}{t + \frac{1}{t + \frac{2}{t + \frac{3}{t + \frac{4}{...}}}}} , t > 0,  (3)

for the function R(t) := Φ̄(t)/φ(t), where φ(t) is the standard normal density and Φ̄(t) = ∫_{t}^{∞} φ(s)ds is the tail of the standard normal distribution. The function, R(t), is often called the Mills ratio, after John Mills who tabulated it [13] in 1929. The Mills ratio appears in the probability theory, [5, 6, 12, 15] in the context of asymptotic expansions for the normal probability integral, in statistical analysis [8, 13] and in the area of numerical analysis [3, 9, 14, 16] where the functional inequalities and irrational approximations for R(t) are discussed.

In this paper we study the matrices 𝔭 and 𝔮. In Section 2, we show that certain integer sequences in [17] are encapsulated in these matrices. In particular, the triangular array \underline{A180048} is described by (2). In Section 2, we also describe connections between the matrix 𝔭 and the coefficients of the Hermite and Bessel polynomials [1, 2].

In Section 3, we introduce and study the generating polynomials. Despite the fact that the recurrence relations for the polynomials P_{k} and Q_{k} have been known for a very long time^{1}, their coefficients were not systematically studied until recently. The first analysis of these coefficients, published in [9], appeared only in 2006, to the best of our knowledge^{2}.

The statements in the first three sections are elementary. Their proofs are left to the reader. In Section 4, we derive the master equation linking together the generating functions of the Laplace polynomials and the Laplace continued fraction, L(t). In Section 5, the master equation is used for a short derivation of the famous identity discovered by Ramanujan:

\frac{1}{1 + \frac{2}{1 + \frac{3}{...}}} = −1 + \frac{1}{\sqrt{\frac{eπ}{2}} − ∑_{n=0}^{∞} \frac{1}{(2n+1)!!}} .

1Some of these relations were derived by Jacobi in [7].

2Our definition of the coefficients of the polynomials Q_{k}(t) is different from that in [9].

2 Matrices 𝔭 and 𝔮

We shall start with the following recurrence equations for the elements of 𝔭 and 𝔮. Proposition 1. The elements of the matrices 𝔭 and 𝔮 satisfy

m · p_{k,m} = k · p_{k−1,m−1}, k ≥ 1, 1 ≤ m ≤ k, (4)
p_{k+1,m} = p_{k,m−1} + (m + 1) · p_{k,m+1}, k ≥ 0, 1 ≤ m ≤ k,  (5)
q_{k,m} = p_{k,m} + (m + 1) · q_{k−1,m+1}, k ≥ 1, 1 ≤ m ≤ k,  (6)
q_{k,m} = q_{k−1,m−1} + k · q_{k−2,m}, k ≥ 2, 1 ≤ m ≤ k. (7)

Proof. The proof is based on the following lemma. Lemma 2. Write

S_{i}^{(j)}(k) = ∑_{l=i}^{j} \binom{k}{l}, k > j.

Then

S_{0}^{(n−1)}(k) − 2S_{0}^{(n−2)}(k − 1) = \binom{k − 1}{n − 1}.  (8)

The derivation of (4) is straightforward. The derivation of (5), (6) and (7) is based on Lemma 2, which in the case of (7) is used twice.

The elements in the first nine rows and columns of the matrix 𝔭 are given in Table 1.

k  m
0 1 2 3 4 5 6 7 8
0 1 0 0 …  0
1 0 1 0 …  0
2 1 0 1 0 …  0
3 0 3 0 1 0 …  0
4 3 0 6 0 1 0 …  0
5 0 15 0 10 0 1 0 … 0
6 15 0 45 0 15 0 1 0 0
7 0 105 0 105 0 21 0 1 0
8 105 0 420 0 210 0 28 0 1
Table 1: Matrix 𝔭 = ∥p_{k,m}∥.

The matrix 𝔭 encapsulates many remarkable integer sequences. The rows of the matrix describe the coefficients of the Laplace polynomials [10], P_{k}(t) = ∑_{m=0}^{k} p_{k,m}t^{m}. These polynomials will be discussed in the next section. The columns of 𝔭 are integer sequences that can be found in [17]. In particular, the first column is p_{2n,0} = (2n − 1)!!, (n = 1, 2, …), which is \underline{A001147}. In the case m = 1, the column p_{2n−1,1} = p_{2n,0}. If m = 2 then p_{2n+2,2} = (n + 1) · (2n + 1)!!, which is \underline{A001879}. In the case m = 3 we have the sequence \underline{A000457}.

The diagonals of 𝔭 also represent some remarkable integer sequences. The diagonal p_{k,2N−k} represents the coefficients of the Nth Bessel polynomial. Indeed, the Bessel polynomials are

y_{N}(t) = ∑_{j=0}^{N} \frac{(N + j)!}{(N − j)! · j!}(\frac{t}{2})^{j}, N = 0, 1, ….

Their coefficients, B_{N,j} := (N + j)!/((N − j)!j!2^{j}), are equal to p_{N+j,N−j}, for j = 0, 1, …, N.

Thus the Bessel polynomials can be written as y_{N}(t) = ∑_{j=0}^{N} p_{N+j,N−j}t^{j}.

The diagonals k − m = 2n also represent some well-known sequences. If n = 0 then we have the sequence \underline{A000012}. If n = 1, p_{m+2,m} = (m+1)(m+2)/2 is the sequence of triangular numbers \underline{A000217}. If n = 2, p_{m+4,m} is the sequence of tritriangular numbers \underline{A050534}.

Analytical properties of the matrix 𝔮 are equally interesting. The elements in the first eight rows of the matrix 𝔮 are given in Table 2. For small m, Formula (2) can be simplified.

k  m
0 1 2 3 4 5 6 7 8
0 1 0 0 …  0
1 0 1 0 …  0
2 2 0 1 0 …  0
3 0 5 0 1 0 …  0
4 8 0 9 0 1 0 …  0
5 0 33 0 14 0 1 0 … 0
6 48 0 87 0 20 0 1 0 0
7 0 279 0 185 0 27 0 1 0

Table 2: Elements of the matrix 𝔮 = ∥q_{k,m}∥.

Proposition 3. For n = 0, 1, 2, …, we have

q_{2n,0} = (2n)!!,  (9)
q_{2n+1,1} = (2n + 2)!! − (2n + 1)!!,  (10)
q_{2n+2,2} = \frac{1}{2}(2n + 4)!! − (2n + 3)!!,  (11)
q_{2n+3,3} = \frac{1}{3!}(2n + 6)!! − \frac{1}{2!}(2n + 5)!! + \frac{1}{3!}(2n + 3)!!,  (12)
q_{2n+4,4} = \frac{1}{4!}(2n + 8)!! − \frac{1}{3!}(2n + 7)!! + \frac{1}{3!}(2n + 5)!!.  (13)

According to Proposition 3, the first column of the matrix 𝔮, described by (9), corresponds to the sequence \underline{A000165}. The second column, satisfying Equation (10), describes the sequence \underline{A129890}. The third column, satisfying Equation (11), represents the integer sequence \underline{A035101}. This sequence is related to the Catalan numbers (see [17]) but our formula (11) looks simpler. The sequence q_{2n+3,3} represents the integer sequence \underline{A263384}. The sequence q_{2n+4,4}, described by (13), is less known.

The elements of the matrix 𝔮 = ∥q_{k,m}∥ can be represented as a linear combination of the diagonal elements of the matrix 𝔭. Let k ≡ m (mod 2), and n = (k − m)/2. Then [10]

m! · q_{k,m} = ∑_{j=0}^{n}(m + j)! · p_{k−j,m+j}.

3 Polynomials P_{k}(t) and Q_{k}(t)

3.1 Laplace and Jacobi polynomials

Consider the polynomials P_{k}(t) := ∑_{m=0}^{k} p_{k,m}t^{m}, and Q_{k}(t) := ∑_{m=0}^{k} q_{k,m}t^{m}, where p_{k,m} and

q_{k,m} are defined by (1) and (2), respectively. In this section, we shall connect these polynomials with the convergents of the Laplace continued fraction, L(t). For this reason, we call P_{k}(t) and Q_{k}(t) the Laplace polynomials in what follows.

Proposition 4 ([7, 9, 10, 14]). For k ≥ 1, the Laplace polynomials satisfy the following recurrent equations

P_{k+1}(t) = tP_{k}(t) + P′_{k}(t),  (14)
Q_{k}(t) = P_{k}(t) + Q′_{k−1}(t),  (15)
P_{k+1}(t) = tP_{k}(t) + kP_{k−1}(t),  (16)
Q_{k+1}(t) = tQ_{k}(t) + (k + 1)Q_{k−1}(t),  (17)

where P_{0}(t) = Q_{0}(t) = 1.

Proposition 4 allows one to find the polynomials P_{k}(t) and Q_{k}(t) for any integer k ≥ 1. The first eight polynomials are given in Table 3.

\begin{matrix}k & P_{k}(t) & Q_{k−1}(t) \\ 1 & t & 1 \\ 2 & t^{2} + 1 & t \\ 3 & t^{3} + 3t & t^{2} + 2 \\ 4 & t^{4} + 6t^{2} + 3 & t^{3} + 5t \\ 5 & t^{5} + 10t^{3} + 15t & t^{4} + 9t^{2} + 8 \\ 6 & t^{6} + 15t^{4} + 45t^{2} + 15 & t^{5} + 14t^{3} + 33t \\ 7 & t^{7} + 21t^{5} + 105t^{3} + 105t & t^{6} + 20t^{4} + 87t^{2} + 48 \\ 8 & t^{8} + 28t^{6} + 210t^{4} + 420t^{2} + 105 & t^{7} + 27t^{5} + 185t^{3} + 279t \end{matrix}

Table 3: Laplace polynomials P_{k}(t) and Q_{k−1}(t).

For t > 0, we consider the rational functions L_{k}(t) = Q_{k−1}(t)/P_{k}(t), (k = 1, 2, …). The function L_{k}(t) is the kth convergent of the continued fraction^{3}, L(t):

lim_{k→∞} L_{k}(t) = L(t) = R(t), t > 0.

The function R(t) satisfies the differential equation

\frac{dR(t)}{dt} = t · R(t) − 1.  (18)

Let us now consider the derivatives of the function R(t). From (18) we derive

\frac{d^{k}R(t)}{dt^{k}} = t · \frac{d^{k−1}R(t)}{dt^{k−1}} + (k − 1) · \frac{d^{k−2}R(t)}{dt^{k−2}}. k = 2, 3, ….  (19)

It was proved in [7, 9, 10] that ^{4}

\frac{d^{k}R(t)}{dt^{k}} = R(t) · P_{k}(t) − Q_{k−1}(t),  k = 1, 2, ….  (20)

3.2 Laplace and Hermite polynomials

The polynomials P_{k}(t) are closely connected to the Hermite polynomials [1, 2]. Denote the differentiation operator by 𝔇: 𝔇g(t) = \frac{dg(t)}{dt}. Then, as usual, 𝔇^{n}g(t) = \frac{d^{n}g(t)}{dt^{n}}, (n = 1, 2, …). Recall that the Hermite polynomials can be defined as

H_{k}(t) := (−1)^{k}e^{t^{2}/2} 𝔇^{k}e^{−t^{2}/2}.

3See [9] and [10] regarding this statement

4Equation (20) follows from (18), (19) and Proposition 4.

Lemma 5. The Laplace polynomials, P_{k}(t), satisfy

P_{k}(t) = e^{−t^{2}/2} 𝔇^{k}e^{t^{2}/2}, k = 0, 1, 2, ….  (21)

Proof. We shall prove this lemma by induction. For k = 1

e^{−t^{2}/2} 𝔇e^{t^{2}/2} = t = P_{1}(t).

Suppose that e^{−t^{2}/2} 𝔇^{n}e^{t^{2}/2} = P_{n}(t). Let us verify that

P_{n+1}(t) = e^{−t^{2}/2} 𝔇^{n+1}e^{t^{2}/2}.

Indeed,

𝔇^{n+1}e^{t^{2}/2} = 𝔇(𝔇^{n}e^{t^{2}/2})
= 𝔇(e^{t^{2}/2} · P_{n}(t))
= e^{t^{2}/2} (tP_{n}(t) + P′_{n}(t)).

Therefore

e^{−t^{2}/2} 𝔇^{n+1}e^{t^{2}/2} = tP_{n}(t) + P′_{n}(t).

Since Equation (14) and the initial condition, P_{1}(t) = t, uniquely determine the sequence of polynomials generated by the recurrence (14), we obtain

e^{−t^{2}/2} 𝔇^{n+1}e^{t^{2}/2} = P_{n+1}(t),

as was to be proved.

Lemma 5 implies (see [9, 10])

P_{k}(t) = (−i)^{k}H_{k}(it).  (22)

This relation allows us to reformulate the classical results obtained for the Hermite polynomials in terms of the Laplace polynomials. In particular, one can easily derive from (22) and the generating function of the Hermite polynomials,

H(t, s) := ∑_{k=0}^{∞} H_{k}(t)\frac{s^{k}}{k!} = e^{st−s^{2}/2} ,

the generating function P(s, t) = ∑_{k=0}^{∞} P_{k}(t)\frac{s^{k}}{k!} .

Lemma 6 ([9, 10]). The generating function, P(s, t), is

P(s, t) = exp(st + \frac{s^{2}}{2} ).  (23)

4 Master equation

Let us now compute the generating function of the Laplace polynomials Q_{k}(t). Write

Q(s, t) := ∑_{k=0}^{∞} ∑_{m=0}^{∞} q_{k,m}t^{m} \frac{s^{k+1}}{(k + 1)!}.

Lemma 7. The generating function Q(s, t) is

Q(s, t) = \sqrt{2π} e^{(s+t)^{2}/2} · (Φ̄(t) − Φ̄(s + t)).  (24)

Proof. We have,

Q(s, t) = ∑_{k=0}^{∞} Q_{k}(t)\frac{s^{k+1}}{(k + 1)!} and P(s, t) = ∑_{k=0}^{∞} P_{k}(t)\frac{s^{k}}{k!} .

The Taylor series expansion for the function R(t) can be written as

R(s + t) = ∑_{k=0}^{∞} \frac{d^{k}R(t)}{dt^{k}} \frac{s^{k}}{k!} .

Recall that the derivatives of the function R(t) satisfy Equation (20):

\frac{d^{k}R(t)}{dt^{k}} = P_{k}(t)R(t) − Q_{k−1}(t).

We have

R(s + t) = \sqrt{2π}e^{(s+t)^{2}/2}Φ̄(s + t) .

Therefore

\sqrt{2π} e^{(s+t)^{2}/2} · Φ̄(s + t) = ∑_{k=0}^{∞} \frac{d^{k}R(t)}{dt^{k}} \frac{s^{k}}{k!}
= R(t) ∑_{k=0}^{∞} P_{k}(t)\frac{s^{k}}{k!} − ∑_{k=0}^{∞} Q_{k−1}(t)\frac{s^{k}}{k!}
= R(t) · e^{st+s^{2}/2} − Q(s, t)
= \sqrt{2π} e^{t^{2}/2} · Φ̄(t) · e^{st+s^{2}/2} − Q(s, t).

Finally, we obtain

Q(s, t) = \sqrt{2π} e^{(s+t)^{2}/2} · (Φ̄(t) − Φ̄(s + t)).

Equation (24) is thus proved.

Theorem 8. The generating functions, P(s, t) and Q(s, t), satisfy

R(s + t) + Q(s, t) = P(s, t)R(t).  (25)

Proof. We have \sqrt{2π} exp(t^{2}/2) · Φ̄(t) = R(t). From Lemma 7 we derive

Q(s, t) = \sqrt{2π} e^{(s+t)^{2}/2} · (Φ̄(t) − Φ̄(s + t))
= e^{st+s^{2}/2} · R(t) − R(t + s).

The latter relation is equivalent to (25).

5 Identities

In [10] we derived a new, to the best of our knowledge, identity

∑_{n=0}^{∞} ∑_{m=0}^{∞} ∑_{j=0}^{n} \frac{(n + m)!}{m!j!(m + 2n + 1 − j)!} 2^{−n} = \sqrt{2π}e^{2}(Φ(2) − Φ(1)),

having an unusual combination of constants in the right hand side. In fact, the whole derivation is based on the substitution s = t = 1 into (25) in Theorem 8. Choosing integer values for s and t we obtain a series of elegant combinatorial identities similar to those discussed in [10]. Another interesting, unexpected connection to the Laplace continued fraction is the Ramanujan identity [4, Entry 43, p. 166]:

\frac{1}{1 + \frac{2}{1 + \frac{3}{1 + \frac{4}{1 + \frac{5}{...}}}}} = \frac{1}{\sqrt{\frac{eπ}{2}} − ∑_{n=0}^{∞} \frac{1}{(2n+1)!!}} − 1.  (26)

This identity can be derived from Equation (25). We will need the following generalization of (26).

Proposition 9. For s > 0, consider the continued fraction

S(s) = \frac{1}{s + \frac{2}{s + \frac{3}{s + \frac{4}{s + \frac{5}{...}}}}} .

Then

S(s) = \frac{1}{e^{s^{2}/2}\sqrt{\frac{π}{2}} − ∑_{n=0}^{∞} \frac{s^{2n+1}}{(2n+1)!!}} − s.  (27)

Proof. From (25) we derive

R(s) + Q(s, 0) = e^{s^{2}/2} · R(0),  (28)

and

Q(s, 0) = ∑_{n=0}^{∞} q_{2n,0} \frac{s^{2n+1}}{(2n + 1)!}
= ∑_{n=0}^{∞}(2n)!!\frac{s^{2n+1}}{(2n + 1)!}
= ∑_{n=0}^{∞} \frac{s^{2n+1}}{(2n + 1)!!}.  (29)

Inspecting the continued fractions (3) and S(s) we find

S(s) = −s + \frac{1}{L(s)},  (30)

and, therefore, for s > 0

S(s) = −s + \frac{1}{R(s)}.

Taking into account that R(0) = \sqrt{π/2} we obtain from the latter equation, (28) and (29)

\frac{1}{s + S(s)} = e^{s^{2}/2}\sqrt{\frac{π}{2}} − ∑_{n=0}^{∞} \frac{s^{2n+1}}{(2n + 1)!!}.  (31)

Equation (31) is equivalent to (27).

Substituting s = 1 into (27) we obtain the Ramanujan identity (26). Remark 10. The series ∑_{n=0}^{∞} \frac{s^{2n+1}}{(2n+1)!!} converges to the function

e^{s^{2}/2} ∫_{0}^{s} e^{−u^{2}/2}du = (Φ(s) − 0.5)/φ(s),

where φ(s) = exp(−s^{2}/2)/\sqrt{2π}. The coefficients of the convergents of the continued fraction S(s) are described by the triangular array \underline{A180048}.

6 Acknowledgments

I am very grateful to Ian Iscoe, Sebastian Jaimungal, Alexey Kuznetsov, Tom Salisbury, Eugene Seneta, Isaac Sonin, Hans Tuenter, Alek Vainshtein, Vladimir Vinogradov, and Vladimir Zhuravlev for interesting comments and stimulating discussions. I would also like to thank the referee for the suggestions and critical comments. This paper would never appear without the OEIS.

References

[1] M. Abramovitz and I. Stegun, Handbook of Mathematical Functions, Dover Publications, New York, 1970.

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

[3] A. Baricz, Mills’ ratio: Monotonicity patterns and functional inequalities, J. Math. Anal. Appl. 340 (2008), 1362–1370.

[4] B. C. Berndt, Ramanujan’s Notebooks, Vol. 2, Springer-Verlag, 1989.

[5] L. Dűembgen, Bounding standard Gaussian tail probability. University of Bern, Institute of Mathematical Statistics and Actuarial Science, Technical Report 76, 2010. Available

at https://arxiv.org/abs/1012.2063.

[6] W. Feller, An Introduction to Probability Theory and its Applications, 1, John Wiley and Sons, New York, 1970.

[7] C. G. J. Jacobi, De fractione continua, in quam integrale, ∫_{x}^{∞} e^{−xx}dx evolvere licet, J. Reine Angew. Math. 12 (1834), 346–347.

[8] M. Kendall and A. Stuart, The Advanced Theory of Statistics, 1, Wiley, 6th edition, 1994.

[9] O. Kouba, Inequalities related to the error function, preprint,

http://arXiv:math/0607694, 2006.

[10] A. Kreinin,  Combinatorial properties of Mills’ ratio,  preprint,

http://arXiv.org/abs/1405.5852, 2014.

[11] P. S. Laplace, Traité de Mécanique Celeste, Vol. 4, Courcier, 1805.

[12] H. McKean, Stochastic Integrals, Academic Press, 1969.

[13] J. Mills, Table of the ratio area to bounding ordinate for any portion of normal curve, Biometrika 18 (1926), 395–400.

[14] I. Pinelis, Monotonicity properties of the relative error of a Padé approximation for Mills’ ratio, J. Inequal. Pure Appl. Math. 3 (2002), 1–8.

[15] H. Ruben, A new asymptotic expansion for the normal probability integral and Mill’s [sic] ratio. J. R. Statist. Soc. 24 (1952), 177–179.

[16] L. Shenton, Inequalities for the normal integral including a new continued fraction, Biometrika, 41 (1954), 177–189.

[17] N. J. A. Sloane, Online Encyclopedia of Integer Sequences. Published electronically at

    https://oeis.org.

2000 Mathematics Subject Classification: Primary 11Y05; Secondary 11Y55. Keywords: continued fraction, integer sequence, Ramanujan identity.

(Concerned with sequences \underline{A000012}, \underline{A000165}, \underline{A000217}, \underline{A000457}, \underline{A001147}, \underline{A001879}, \underline{A035101}, \underline{A050534}, \underline{A129890}, \underline{A180048}, and \underline{A263384}.)

Received November 11 2015; revised versions received April 5 2016; June 24 2016; June 27 2016. Published in Journal of Integer Sequences, June 27 2016.

Return to Journal of Integer Sequences home page.