A New Approach to Multivariate q-Euler Polynomials Using the Umbral Calculus

Serkan Araci Atatürk Street 31290 Hatay

Turkey

mtsrkn@hotmail.com

Xiangxing Kong

Department of Mathematics and Statistics

Central South University

Changsha 410075

China

xiangxingkong@gmail.com

Mehmet Acikgoz Department of Mathematics

University of Gaziantep

27310 Gaziantep

Turkey

acikgoz@gantep.edu.tr

Erdoğan Şen Department of Mathematics Namik Kemal University

59030 Tekirdağ

Turkey

erdogan.math@gmail.com

Abstract

We derive numerous identities for multivariate q-Euler polynomials by using the umbral calculus.

1 Preliminaries

Throughout this paper, we use the following notation, where ℂ denotes the set of complex numbers, F denotes the set of all formal power series in the variable t over ℂ with F = \{f(t) = ∑_{k=0}^{∞} a_{k} \frac{t^{k}}{k!} | a_{k} ∈ ℂ\}, P = ℂ[x] and P^{∗} denotes the vector space of all linear functional on P, ⟨L | p(x)⟩ denotes the action of the linear functional L on the polynomial p(x), and it is well-known that the vector space operation on P^{∗} is defined by

⟨L + M | p(x)⟩ = ⟨L | p(x)⟩ + ⟨M | p(x)⟩ ,
⟨cL | p(x)⟩ = c ⟨L | p(x)⟩ ,

where c is some constant in ℂ (for details, see [5, 6, 8, 11]).

The formal power series are known by the rule

f(t) = ∑_{k=0}^{∞} a_{k} \frac{t^{k}}{k!} ∈ F

which defines a linear functional on P as ⟨f(t) | x^{n}⟩ = a_{n} for all n ≥ 0 (for details, see [5, 6, 8, 11]]). Additionally,

⟨t^{k} | x^{n}⟩ = n!δ_{n,k},  (1)

where δ_{n,k} is the Kronecker symbol. When we take f_{L}(t) = ∑_{k=0}^{∞} ⟨L | x^{k}⟩ \frac{t^{k}}{k!}, then we obtain ⟨f_{L}(t) | x^{n}⟩ = ⟨L | x^{n}⟩ and so as linear functionals L = f_{L}(t) (see [5, 6, 8, 11]). Additionally, the map L → f_{L}(t) is a vector space isomorphism from P^{∗} onto F. Henceforth, F denotes both the algebra of the formal power series in t and the vector space of all linear functionals on P, and so an element f(t) of F can be thought of as both a formal power series and a linear functional. The algebra F is called the umbral algebra (see [5, 6, 8, 11]).

Also, the evaluation functional for y in ℂ is defined to be power series e^{yt}. We can write that ⟨e^{yt} | x^{n}⟩ = y^{n} and so ⟨e^{yt} | p(x)⟩ = p(y) (see [5, 6, 8, 11]). We note that for all f(t) in F

f(t) = ∑_{k=0}^{∞} ⟨f(t) | x^{k}⟩ \frac{t^{k}}{k!}  (2)

and for all polynomials p(x),

p(x) = ∑_{k=0}^{∞} ⟨t^{k} | p(x)⟩ \frac{x^{k}}{k!} ,  (3)

(for details, see [5, 6, 8, 11]). The order o(f(t)) of the power series f(t) ≠ 0 is the smallest integer k for which a_{k} does not vanish. It is considered o(f(t)) = ∞ if f(t) = 0. We see that o(f(t)g(t)) = o(f(t)) + o(g(t)) and o(f(t) + g(t)) ≥ min {o(f(t)), o(g(t))}. The series f(t) has a multiplicative inverse, denoted by f(t)^{−1} or \frac{1}{f(t)}, if and only if o(f(t)) = 0. Such series is called an invertible series. A series f(t) for which o(f(t)) = 1 is called a delta series (see [5, 6, 8, 11]). For f(t), g(t) ∈ F, we have ⟨f(t)g(t) | p(x)⟩ = ⟨f(t) | g(t)p(x)⟩.

A delta series f(t) has a compositional inverse \overline{f}(t) such that f (\overline{f}(t)) = \overline{f} (f(t)) = t. For f(t), g(t) ∈ F , we have ⟨f(t)g(t) | p(x)⟩ = ⟨f(t) | g(t)p(x)⟩. By (3), we have

p^{(k)} (x) = \frac{d^{k}p(x)}{dx^{k}} = ∑_{l=k}^{∞} \frac{⟨t^{l} | p(x)⟩}{l!}l (l − 1) ⋯ (l − k + 1) x^{l−k}.  (4)

Thus, we see that

p^{(k)} (0) = ⟨t^{k} | p(x)⟩ = ⟨1 | p^{(k)} (x)⟩.  (5)

By (4), we get

t^{k}p(x) = p^{(k)} (x) = \frac{d^{k}p(x)}{dx^{k}}.  (6)

So, we have

e^{yt}p(x) = p (x + y) .  (7)

Let f(t) be a delta series and let g(t) be an invertible series. Then there exists a unique sequence S_{n}(x) of polynomials, with deg S_{n}(x) = n, such that ⟨g(t)f(t)^{k} | S_{n}(x)⟩ = n!δ_{n,k} for all n, k ≥ 0. The sequence S_{n}(x) is called the Sheffer sequence for (g(t), f(t)) or that S_{n}(t) is Sheffer for (g(t), f(t)).

The Sheffer sequence for (1, f(t)) is called the associated sequence for f(t); we also say S_{n}(x) is associated with f(t). The Sheffer sequence for (g(t), t) is called the Appell sequence for g(t); we also say S_{n}(x) is Appell for g(t).

Let p(x) ∈ P. Then we have

⟨\frac{e^{yt} − 1}{t} | p(x)⟩ = ∫_{0}^{y} p(u)du,
⟨f(t) | xp(x)⟩ = ⟨∂_{t}f(t) | p(x)⟩ = ⟨f´ (t) | p(x)⟩ ,  (8)
⟨e^{yt} − 1 | p(x)⟩ = p(y) − p(0), (see [5, 6, 8, 11]).

Let S_{n} (x) be Sheffer for (g (t) , f (t)). Then the following results are known in [11]:

h (t) = ∑_{k=0}^{∞} \frac{⟨h (t) | S_{k} (x)⟩}{k!}g (t) f (t)^{k} , h (t) ∈ F

p(x) = ∑_{k=0}^{∞} \frac{⟨g (t) f(t)^{k} | p(x)⟩}{k!}S_{k} (x) , p(x) ∈ P,

\frac{1}{g (\overline{f} (t))}e^{y\overline{f}(t)} = ∑_{k=0}^{∞} S_{k} (y) \frac{t^{k}}{k!}, for all y ∈ ℂ,  (9)
f(t)S_{n} (x) = nS_{n−1}(x).

Let a_{1}, …, a_{r}, b_{1}, …, b_{r} be positive integers. Kim and Rim [1] defined the generating function for multivariate q-Euler polynomials as follows:

F_{q}(t, x | a_{1}, …, a_{r}; b_{1}, …, b_{r}) = ∑_{n=0}^{∞} E_{n,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r}) \frac{t^{n}}{n!}  (10)

= \frac{2^{r}}{(q^{b_{1}}e^{a_{1}t} + 1) ⋯ (q^{b_{r}}e^{a_{r}t} + 1)}e^{xt}.

Note that

E_{0,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r}) = \frac{2^{r}}{[2]_{q^{b_{1}}} [2]_{q^{b_{2}}} ⋯ [2]_{q^{br}}} ,

where [x]_{q} is q-extension of x defined by

[x]_{q} = \frac{q^{x} − 1}{q − 1} = 1 + q + q^{2} + ⋯ + q^{x−1}.

We assume that q ∈ ℂ with |q| < 1. Also, we note that lim_{q→1} [x]_{q} = x (see [1]–[11]). In the special case, x = 0, E_{n,q} (0 | a_{1}, …, a_{r}; b_{1}, …, b_{r}) := E_{n,q} (a_{1}, …, a_{r}; b_{1}, …, b_{r}) are called multivariate q-Euler numbers. By (10), we obtain the following:

E_{n,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r}) = ∑_{k=0}^{n} \binom{n}{k}x^{k}E_{n−k,q} (a_{1}, …, a_{r}; b_{1}, …, b_{r}) .  (11)

Kim and Kim [5] studied some interesting identities for Frobenius-Euler polynomials arising from umbral calculus. They derived not only new but also fascinating identities in modern classical umbral calculus.

By the same motivation, we also get numerous identities for multivariate q-Euler polynomials by utilizing from the umbral calculus.

2 On the multivariate q-Euler polynomials arising from umbral calculus

Assume that S_{n} (x) is an Appell sequence for g (t). By (9), we have

\frac{1}{g(t)}x^{n} = S_{n}(x) if and only if x^{n} = g(t)S_{n}(x), (n ≥ 0) .  (12)

Let us take

g (t | a_{1}, …, a_{r}; b_{1}, …, b_{r}) = \frac{(q^{b_{1}}e^{a_{1}t} + 1) ⋯ (q^{b_{r}}e^{a_{r}t} + 1)}{2^{r}} ∈ F.

Then we readily see that g (t | a_{1}, …, a_{r}; b_{1}, …, b_{r}) is an invertible series. By (12), we have

∑_{n=0}^{∞} E_{n,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r}) \frac{t^{n}}{n!} = \frac{1}{g (t | a_{1}, …, a_{r}; b_{1}, …, b_{r})}e^{xt}.  (13)

By (13), we obtain the following

\frac{1}{g (t | a_{1}, …, a_{r}; b_{1}, …, b_{r})}x^{n} = E_{n,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r}) .  (14)

Also, by (6), we have

tE_{n,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r}) = E_{´ n,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r})  (15)
= nE_{n−1,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r}) .

By (14) and (15), we have the following proposition.

Proposition 1. For n ≥ 0, E_{n,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r}) is an Appell sequence for

g (t | a_{1}, …, a_{r}; b_{1}, …, b_{r}) = \frac{(q^{b_{1}}e^{a_{1}t} + 1) ⋯ (q^{b_{r}}e^{a_{r}t} + 1)}{2^{r}}.

By (10), we see that

∑_{n=1}^{∞} E_{n+1,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r}) \frac{t^{n}}{n!} = \frac{xge^{xt} − gé^{xt}}{g^{2}}  (16)

= ∑_{n=0}^{∞} (x\frac{1}{g}x^{n} − \frac{g}{ǵ}\frac{1}{g}x^{n}) \frac{t^{n}}{n!}

where we used g := g (t | a_{1}, …, a_{r}; b_{1}, …, b_{r}). Because of (14) and (16), we discover the following:

E_{n+1,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r})  (17)

= xE_{n,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r}) − \frac{g}{ǵ}E_{n,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r}) .

Therefore, we deduce the following theorem.

Theorem 2. Let g := g (t | a_{1}, …, a_{r}; b_{1}, …, b_{r}) = \frac{(q^{b_{1}}e^{a_{1}t}+1)···(q^{br} e^{art}+1)}{2^{r}} ∈ F. Then we have for n ≥ 0 :

E_{n+1,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r}) = (x − \frac{g}{ǵ}) E_{n,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r}) .  (18)

From (10), we derive that

∑_{n=0}^{∞} (q^{b_{r}}E_{n,q} (x + a_{r} | a_{1}, …, a_{r}; b_{1}, …, b_{r}) + E_{n,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r})) \frac{t^{n}}{n!}  (19)

= 2 ∑_{n=0}^{∞} E_{n,q} (x | a_{1}, …, a_{r−1}; b_{1}, …, b_{r−1}) \frac{t^{n}}{n!}.

By comparing the coefficients in the both sides of \frac{t^{n}}{n!} on the above, we obtain the following

2E_{n,q} (x | a_{1}, …, a_{r−1}; b_{1}, …, b_{r−1}) = q^{b_{r}}E_{n,q} (x + a_{r} | a_{1}, …, a_{r}; b_{1}, …, b_{r})  (20)
+E_{n,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r}) .

From Theorem 2, we get the following equation

gE_{n+1,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r})  (21)
= gxE_{n,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r}) − gÉ_{n,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r}) .

By using (20) and (21), we arrive at the desired theorem. Theorem 3. For n ≥ 0, we have

2E_{n,q} (x | a_{1}, …, a_{r−1}; b_{1}, …, b_{r−1}) = q^{b_{r}}E_{n,q} (x + a_{r} | a_{1}, …, a_{r}; b_{1}, …, b_{r})  (22)
+E_{n,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r}) .

Now, we consider that

∫_{x}^{x+y} E_{n,q} (u | a_{1}, …, a_{r}; b_{1}, …, b_{r}) du
= \frac{1}{n + 1} (E_{n+1,q} (x + y | a_{1}, …, a_{r}; b_{1}, …, b_{r}) − E_{n+1,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r}))
= \frac{1}{n + 1} ∑_{j=1}^{∞} \binom{n + 1}{j}E_{n+1−j,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r}) y^{j}
= ∑_{j=1}^{∞} \frac{n (n − 1) (n − 2) ⋯ (n − j + 2)}{j!}E_{n+1−j,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r}) y^{j}
= \frac{1}{t} (∑_{j=0}^{∞} \frac{y^{j}t^{j}}{j!} − 1) E_{n,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r})
= \frac{e^{yt} − 1}{t}E_{n,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r}) .

Therefore, we discover the following theorem:

Theorem 4. For n ≥ 0, we have

∫_{x}^{x+y} E_{n,q} (u | a_{1}, …, a_{r}; b_{1}, …, b_{r}) du = \frac{e^{yt} − 1}{t}E_{n,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r}) .  (23)

By (15) and Proposition 1, we have

t \{\frac{1}{n + 1}E_{n+1,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r})\} = E_{n,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r}) . (24)

Thanks to (24), we readily derive the following:

⟨e^{yt} − 1 | \frac{E_{n+1,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r})}{n + 1}⟩  (25)
= ⟨\frac{e^{yt} − 1}{t} | t \{\frac{E_{n+1,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r})}{n + 1}\}⟩
= ⟨\frac{e^{yt} − 1}{t} | E_{n,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r})⟩ .

On account of (8) and (24), we get

⟨\frac{e^{yt} − 1}{t} | E_{n,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r})⟩ = ⟨e^{yt} − 1 | \frac{E_{n+1,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r})}{n + 1}⟩

= \frac{1}{n + 1} \{E_{n+1,q} (y | a_{1}, …, a_{r}; b_{1}, …, b_{r}) − E_{n+1,q} (a_{1}, …, a_{r}; b_{1}, …, b_{r})\}

= ∫_{0}^{y} E_{n,q} (u | a_{1}, …, a_{r}; b_{1}, …, b_{r}) du.

Consequently, we obtain the following theorem. Theorem 5. For n ≥ 0, we have

⟨\frac{e^{yt} − 1}{t} | E_{n,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r})⟩ = ∫_{0}^{y} E_{n,q} (u | a_{1}, …, a_{r}; b_{1}, …, b_{r}) du. (26)

Assume that P (q | a_{1}, …, a_{r}; b_{1}, …, b_{r}) = {p(x) ∈ Q (q | a_{1}, …, a_{r}; b_{1}, …, b_{r}) [x] | deg p(x) ≤ n}

is a vector space over Q (q | a_{1}, …, a_{r}; b_{1}, …, b_{r}) which are the space of all polynomials including coefficients q, a_{1}, …, a_{r}, b_{1}, …, b_{r}.

For p(x) ∈ P (q | a_{1}, …, a_{r}; b_{1}, …, b_{r}), let us consider

p(x) = ∑_{k=0}^{n} b_{k}E_{k,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r}) .  (27)

By Proposition 1, E_{n,q} (u | a_{1}, …, a_{r}; b_{1}, …, b_{r}) is an Appell sequence for

g := g (t | a_{1}, …, a_{r}; b_{1}, …, b_{r}) = \frac{(q^{b_{1}}e^{a_{1}t} + 1) ⋯ (q^{b_{r}}e^{a_{r}t} + 1)}{2^{r}}.

Thus we have

⟨g (t | a_{1}, …, a_{r}; b_{1}, …, b_{r}) t^{k} | E_{n,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r})⟩ = n!δ_{n,k}.  (28)

From (27) and (28), we compute

⟨g (t | a_{1}, …, a_{r}; b_{1}, …, b_{r}) t^{k} | p(x)⟩ = ∑_{l=0}^{n} b_{l} ⟨gt^{k} | E_{l,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r})⟩ (29)

= ∑_{l=0}^{n} b_{l}l!δ_{l,k} = k!b_{k}.

Thus, by (29), we derive

b_{k} = \frac{1}{k!} ⟨gt^{k} | p(x)⟩  (30)
= \frac{1}{2^{r}k!} ⟨(q^{b_{1}}e^{a_{1}t} + 1) ⋯ (q^{b_{r}}e^{a_{r}t} + 1) | p^{(k)} (x)⟩.

It is not difficult to show the following

(q^{b_{1}}e^{a_{1}t} + 1) ⋯ (q^{b_{r}}e^{a_{r}t} + 1) = ∑_{k_{1},...,k_{r}≥0_{k_{1}+k_{2}+...+kr=1}} q^{∑_{l=1}^{r} b_{l}k_{l}}e^{t ∑_{j=1}^{r} a_{j}k_{j}}.  (31)

Via the results (30) and (31), we easily see that

b_{k} = \frac{1}{2^{r}k!} ∑_{k_{1},...,k_{r}≥0_{k_{1}+k_{2}+...+kr=1}} q^{∑_{l=1}^{r} b_{l}k_{l}} ⟨e^{t ∑_{j=1}^{r} a_{j}k_{j}} | p^{(k)} (x)⟩
= \frac{1}{2^{r}k!} ∑_{k_{1},...,k_{r}≥0_{k_{1}+k_{2}+...+kr=1}} q^{∑_{l=1}^{r} b_{l}k_{l}}p^{(k)} (∑_{j=1}^{r} a_{j}k_{j}).

As a result, we state the following theorem. Theorem 6. For p(x) ∈ P (q | a_{1}, …, a_{r}; b_{1}, …, b_{r}), when we consider

p(x) = ∑_{k=0}^{n} b_{k}E_{k,q} (x | a_{1}, …, a_{r}; b_{1}, …, b_{r}) ,

we obtain

b_{k} = \frac{1}{2^{r}k!} ∑_{k_{1},...,k_{r}≥0_{k_{1}+k_{2}+...+kr=1}} q^{∑_{l=1}^{r} b_{l}k_{l}}p^{(k)} (∑_{j=1}^{r} a_{j}k_{j}).

3 Acknowledgements

The authors would like to thank the anonymous referee for several helpful comments.

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2010 Mathematics Subject Classification: Primary 11S80; Secondary 11B68. Keywords: Appell sequence, Sheffer sequence, multivariate q-Euler polynomial, formal power series.

Received January 23 2013; revised versions received June 5 2013; December 7 2013. Published in Journal of Integer Sequences, December 8 2013.

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