On Generating Functions for Parametrically Generalized Polynomials Involving Combinatorial, Bernoulli and Euler Polynomials and Numbers

Abdelmejid Bayad

^1,*

and

Yilmaz Simsek

Département de Mathématiques, Université d’Evry Val d’Essonne, Bd. F. Mitterrand, CEDEX, 91025 Evry, France

Department of Mathematics, Faculty of Arts and Science, University of Akdeniz, Antalya 07058, Turkey

Author to whom correspondence should be addressed.

Symmetry2022,14(4), 654;https://doi.org/10.3390/sym14040654

Submission received: 8 February 2022 /Revised: 15 March 2022 /Accepted: 22 March 2022 /Published: 23 March 2022

(This article belongs to the Special IssueRecent Advances in Number Theory and Their Applications)

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Abstract

The aim of this paper is to give generating functions for parametrically generalized polynomials that are related to the combinatorial numbers, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the cosine-Bernoulli polynomials, the sine-Bernoulli polynomials, the cosine-Euler polynomials, and the sine-Euler polynomials. We investigate some properties of these generating functions. By applying Euler’s formula to these generating functions, we derive many new and interesting formulas and relations related to these special polynomials and numbers mentioned as above. Some special cases of the results obtained in this article are examined. With this special case, detailed comments and comparisons with previously available results are also provided. Furthermore, we come up with open questions about interpolation functions for these polynomials. The main results of this paper highlight the existing symmetry between numbers and polynomials in a more general framework. These include Bernouilli, Euler, and Catalan polynomials.

Keywords:

Bernoulli and Euler numbers and polynomials;cosine-type Bernoulli and Euler polynomials;sine-type Bernoulli and Euler polynomials;Stirling numbers;generating functions;special numbers and polynomials

MSC:

05A15; 11B68; 11B73; 11B83; 33B10

1. Motivation and Preliminaries

1.1. Motivation

For

k \in Z \ {- 2}

and

a \in C

, in [1], Simsek introduced the functions

F_{Y} (w, k, a) = \frac{a w}{e^{(k + 1) w} - e^{- (k + 1) w} + e^{w} - e^{- w}} = \sum_{n = 0}^{\infty} Y_{n} (k, a) \frac{w^{n}}{n!},

(1)

K_{Y} (w, x, k, a) = e^{w x} F_{Y} (w, k, a) = \sum_{n = 0}^{\infty} Q_{n} (x, k, a) \frac{w^{n}}{n!} .

(2)

Letting

x = 0

in (2), we have

Q_{n} (0, k, a) = Y_{n} (k, a) .

Another interesting example is given by

B_{n} (x) = 2^{- n} Q_{n} (2 x - 1, - 1, 2)

which are the well-known Bernoulli polynomials. Moreover, we have

B_{n} = 2^{- n} \sum_{v = 0}^{n} {(- 1)}^{v} (\binom{n}{v}) Y_{n - v} (- 1, 2)

where

B_{n}

are the Bernoulli numbers.

Bernoulli numbers and polynomials are very important and fundamental in many areas. Bernoulli numbers sit in the center of a number of mathematical fields. Among other things, we can mention, for example that

-: Bernoulli numbers are rational numbers;
-: Their numerators are very important for differential topology via the Kervaire–Milnor Formula;
-: Their denominators are very important for homotopy theory;
-: Bernoulli number are central in Number theory and are special values of zeta functions on integers;
-: Interpolation theory connects Bernoulli Numbers and of Eisenstein series, modular forms, and complex analysis;
-: Homotopy theory and number theory and the special values of zetas functions on the integers.

For more details, see [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22].

Thus, the Bernoulli numbers and polynomials have received much considerable attention throughout the mathematical literature.

Because of the limited attention given to the numbers

Y_{n} (k, a)

and the polynomials

Q_{n} (x, k, a)

, there is no question of a standard. Among other things, in this article, we present a systematic treatment of these polynomials and numbers. This will give new insight into the subject.

By using (1) and (2), it is easy to see that the polynomials

Q_{n} (x, k, a)

and the numbers

Y_{n} (k, a)

are linked by the following relationship:

Q_{n} (x, k, a) = \sum_{v = 0}^{n} (\binom{n}{v}) Y_{n - v} (k, a) x^{v}

(3)

(cf. [1]).

We now introduce the following generating functions involving

sin (y w)

and

cos (y w)

for the following parametrically generalized polynomials: the polynomials

Q_{n}^{(S)} (x, y, k, a)

and

Q_{n}^{(C)} (x, y, k, a)

, respectively:

H (w, x, y, a, k) = \frac{e^{x w} sin (y w) a w}{e^{(k + 1) w} - e^{- (k + 1) w} + e^{w} - e^{- w}} = \sum_{n = 0}^{\infty} Q_{n}^{(S)} (x, y, k, a) \frac{w^{n}}{n!}

(4)

and

G (w, x, y, a, k) = \frac{e^{x w} cos (y w) a w}{e^{(k + 1) w} - e^{- (k + 1) w} + e^{w} - e^{- w}} = \sum_{n = 0}^{\infty} Q_{n}^{(C)} (x, y, k, a) \frac{w^{n}}{n!},

(5)

where

k \in Z \ {- 2}

and

a \in C

We investigate and study these generating functions. InSection 2, by using these generating functions and their functional equations, we give relations among the polynomials

Q_{n}^{(S)} (x, y, k, a)

, the polynomials

Q_{n}^{(C)} (x, y, k, a)

, the polynomials

Q_{n} (x, k, a)

, the numbers

Y_{n - v} (k, a)

, the Bernoulli polynomials and numbers, the Euler polynomials and numbers, the cosine-Bernoulli polynomials, the sine-Bernoulli polynomials, the cosine-Euler polynomials, and the sine-Euler polynomials.

We now give in more detail the contents of this paper. InSection 2, by using generating functions for parametrically generalized polynomials and numbers, we obtain interesting identities and relations including the polynomials

Q_{n}^{(S)} (x, y, k, a)

, the polynomials

Q_{n}^{(C)} (x, y, k, a)

, the polynomials

Q_{n} (x, k, a)

, the numbers

Y_{n - v} (k, a)

β_{n} (k)

, and the Stirling numbers of the second kind.

InSection 3, we give open questions related to the interpolation functions for the polynomials

Q_{n} (x, k, a)

Q_{n}^{(C)} (x, y, k, a)

, and

Q_{n}^{(S)} (x, y, k, a)

InSection 4, we provide a concluding statement.

1.2. Preliminaries

In order to give the results of this paper, we need the following standard notation, definitions, and relations. Throughout this paper, we use the following notations:

N = \{1, 2, \dots\}, N_{0} = \{0, 1, 2, \dots\} = N \cup \{0\} .

As usual,

Z

Q

R_{c}

and

C

denote the set of integers, the set of rational numbers, the set of real numbers, and the set of complex numbers, respectively. Moreover, the falling factorial is given by

{(λ)}_{n} = \{\begin{matrix} λ (λ - 1) (λ - 2) \dots (λ - n + 1) & if n \in N, \\ 1 & if n = 0, \end{matrix}

where

{(λ)}_{0} = 1

and

λ \in C

Let

α \in R

(or

C

). The Bernoulli numbers and polynomials of higher order are defined by means of the following generating functions:

F_{B} (w, α) = {(\frac{w}{e^{w} - 1})}^{α} = \sum_{n = 0}^{\infty} B_{n}^{(α)} \frac{w^{n}}{n!}

(6)

and

G_{B} (w, x, α) = F_{B} (w, α) e^{w x} = \sum_{n = 0}^{\infty} B_{n}^{(α)} (x) \frac{w^{n}}{n!},

(7)

respectively (cf. [18,23,24]).

By substituting

α = 1

into (6) and (7), we obtain the classical Bernoulli numbers and polynomials

B_{n}^{(1)} = B_{n} and B_{n}^{(1)} (x) = B_{n} (x) .

For

α = 0

, we have

B_{n}^{(0)} (x) = x^{n}

and

B_{n}^{(0)} = \{\begin{matrix} 1 & if n = 0, \\ 0 & if n > 0 . \end{matrix}

By using (6) and (7), we have

B_{n}^{(α)} (x) = \sum_{v = 0}^{n} (\binom{n}{v}) x^{v} B_{n - v}^{(α)},

where

n \in N_{0}

(cf. [18,23,24]).

Let

α \in R

(or

C

). The Euler numbers and polynomials of a higher order are defined by means of the following generating functions:

F_{E} (w, α) = {(\frac{2}{e^{w} + 1})}^{α} = \sum_{n = 0}^{\infty} E_{n}^{(α)} \frac{w^{n}}{n!}

(8)

and

G_{E} (w, x, α) = F_{E} (w, α) e^{w x} = \sum_{n = 0}^{\infty} E_{n}^{(α)} (x) \frac{w^{n}}{n!},

(9)

respectively (cf. [18,23,24]).

By substituting

α = 1

into (8) and (9), the classical Euler numbers and polynomials are derived as follows:

E_{n}^{(1)} = E_{n} and E_{n}^{(1)} (x) = E_{n} (x) .

For

α = 0

, we have

E_{n}^{(0)} (x) = x^{n}

and

E_{n}^{(0)} = \{\begin{matrix} 1 & if n = 0, \\ 0 & if n > 0 . \end{matrix}

By using (8) and (9), we have

E_{n}^{(α)} (x) = \sum_{v = 0}^{n} (\binom{n}{v}) x^{v} E_{n - v}^{(α)},

where

n \in N_{0}

(cf. [18,23,24]).

By using (1), (6), and (9), we have the following identity:

Y_{n} (k, a) = \frac{a}{2 (k + 2)} \sum_{v = 0}^{n} (\binom{n}{v}) k^{n - v} {(k + 2)}^{v} B_{v} E_{n - v} (\frac{k + 1}{k}),

(10)

where

n \in N_{0}

(cf. [1]).

By using (1), (7), and (8), we have the following identity:

Y_{n} (k, a) = \frac{a}{2 (k + 2)} \sum_{v = 0}^{n} (\binom{n}{v}) k^{n - v} {(k + 2)}^{v} E_{n - v} B_{v} (\frac{k + 1}{k + 2}),

(11)

where

n \in N_{0}

(cf. [1]).

The Stirling numbers of the second kind,

S_{2} (n, k)

, are defined by means of the following generating function:

F_{S} (w, k) = \frac{{(e^{w} - 1)}^{k}}{k!} = \sum_{n = 0}^{\infty} S_{2} (n, k) \frac{w^{n}}{n!}

(12)

and

y^{n} = \sum_{v = 0}^{n} S_{2} (n, v) {(y)}_{v},

(13)

where

k \in N_{0}

(cf. [18,23,24]).

By using (12), we have

S_{2} (n, k) = \frac{1}{k!} \sum_{j = 0}^{k} {(- 1)}^{k - j} (\binom{k}{j}) j^{n},

where

n, k \in N_{0} .

By using (8) and (12), we get the following identity:

S_{2} (n, k) = \frac{2^{k - n}}{k!} \sum_{m = 0}^{n} \sum_{j = 0}^{k} {(- 1)}^{k - j} (\binom{n}{m}) (\binom{k}{j}) j^{m} E_{n - m}^{(- k)}

(14)

(cf. [23]).

The polynomials

C_{n} (x, y)

and

S_{n} (x, y)

are defined by means of the following generating functions:

K_{C} (w, x, y) = e^{x w} cos (y w) = \sum_{n = 0}^{\infty} C_{n} (x, y) \frac{w^{n}}{n!}

(15)

and

K_{S} (w, x, y) = e^{x w} sin (y w) = \sum_{n = 0}^{\infty} S_{n} (x, y) \frac{w^{n}}{n!},

(16)

(cf. [25,26,27,28,29,30,31]).

The polynomials

C_{n} (x, y)

and

S_{n} (x, y)

are computed by the following formulas

C_{n} (x, y) = \sum_{j = 0}^{[\frac{n}{2}]} {(- 1)}^{j} (\binom{n}{2 j}) x^{n - 2 j} y^{2 j}

and

S_{n} (x, y) = \sum_{j = 0}^{[\frac{n - 1}{2}]} {(- 1)}^{j} (\binom{n}{2 j + 1}) x^{n - 2 j - 1} y^{2 j + 1},

respectively (cf. [25,26,27,28,29,30,31]).

The cosine-Bernoulli polynomials

B_{n}^{(C)} (x, y)

and the sine-Bernoulli polynomials

B_{n}^{(S)} (x, y)

are defined by means of the following generating functions:

g_{C} (w, x, y) = \frac{w cos (y w)}{e^{w} - 1} e^{x w} = \sum_{n = 0}^{\infty} B_{n}^{(C)} (x, y) \frac{w^{n}}{n!}

(17)

and

g_{S} (w, x, y) = \frac{w sin (y w)}{e^{w} - 1} e^{x w} = \sum_{n = 0}^{\infty} B_{n}^{(S)} (x, y) \frac{w^{n}}{n!}

(18)

(cf. [30,31]; see also [26,27,28,29]).

The cosine-Euler polynomials

E_{n}^{(C)} (x, y)

and the sine-Euler polynomials

E_{n}^{(S)} (x, y)

are defined by means of the following generating functions:

h_{C} (w, x, y) = \frac{2 cos (y w)}{e^{w} + 1} e^{x w} = \sum_{n = 0}^{\infty} E_{n}^{(C)} (x, y) \frac{w^{n}}{n!}

(19)

and

h_{S} (w, x, y) = \frac{2 sin (y w)}{e^{w} + 1} e^{x w} = \sum_{n = 0}^{\infty} E_{n}^{(S)} (x, y) \frac{w^{n}}{n!}

(20)

(cf. [30,31]; see also [26,27,28,29]).

Kucukoglu and Simsek [32] defined a new sequence of special numbers

β_{n} (k)

by means of the following generating function:

{(1 - \frac{z}{2})}^{k} = \sum_{n = 0}^{\infty} β_{n} (k) \frac{z^{n}}{n!},

(21)

where

k \in N_{0}

z \in C

with

|z| < 2

By using (21), we have

β_{n} (k) = \frac{{(- 1)}^{n} n!}{2^{n}} (\binom{k}{n}),

(22)

where

n, k \in N_{0}

(cf. [32] (Equation (4.9))).

2. Generating Functions for New Classes of Parametric Kinds of Special Polynomials

In this section, we investigate some properties of Equations (4) and (5). By using these generating functions, we derive some new identities and relations involving the polynomials

Q_{n}^{(S)} (x, y, k, a)

, the polynomials

Q_{n}^{(C)} (x, y, k, a)

, the polynomials

Q_{n} (x, k, a)

, the numbers

Y_{n - v} (k, a)

By substituting

y = 0

and

x = 0

into the Equations (4) and (5), we have the following identities:

Q_{n}^{(C)} (x, 0, k, a) = Q_{n} (x, k, a),

Q_{n}^{(C)} (0, 0, k, a) = Y_{n} (k, a),

and

Q_{n}^{(S)} (x, 0, k, a) = 0 .

In [1], using Equation (1), Simsek gave

F_{Y} (w, k, a) = \frac{a w e^{(k + 1) w}}{(e^{(k + 2) w} - 1) (e^{k w} + 1)} .

Combining the above equation with (7), (9), (17), (18), (19), and (20), we obtain the following functional equations:

\begin{matrix} H (w, x, y, 2, k) & = & \frac{1}{k + 2} g_{S} ((k + 2) w, \frac{k + 1 + x}{k + 2}, \frac{y}{k + 2}) F_{E} (k w, 1) \\ H (w, x, y, 2, k) & = & \frac{1}{k + 2} G_{B} ((k + 2) w, 0, 1) h_{S} (k w, \frac{k + 1 + x}{k}, \frac{y}{k}), \end{matrix}

and

\begin{matrix} G (w, x, y, 2, k) & = & \frac{1}{k + 2} g_{C} ((k + 2) w, \frac{k + 1 + x}{k + 2}, \frac{y}{k + 2}) F_{E} (k w, 1) \\ G (w, x, y, 2, k) & = & \frac{1}{k + 2} G_{B} ((k + 2) w, 0, 1) h_{S} (k w, \frac{k + 1 + x}{k}, \frac{y}{k}) . \end{matrix}

Using similar functional equations to the above equations, we give some novel formulas and relations including the polynomials

Q_{n}^{(S)} (x, y, k, a)

, the polynomials

Q_{n}^{(C)} (x, y, k, a)

, the polynomials

Q_{n} (x, k, a)

, the numbers

Y_{n - v} (k, a)

Theorem 1.

Let

n \in N_{0}

. Then, we have

Q_{n}^{(S)} (x, y, k, a) = \frac{a}{2} \sum_{j = 0}^{n} (\binom{n}{j}) E_{j}^{(S)} (\frac{x}{k}, \frac{y}{k}) B_{n - j} (\frac{k + 1}{k + 2}) k^{j} {(k + 2)}^{n - j - 1} .

Proof.

By using (4), we obtain the following functional equation:

\frac{e^{x w} sin (y w)}{e^{k w} + 1} (\frac{a w}{e^{(k + 2) w} - 1}) e^{(k + 1) w} = \sum_{n = 0}^{\infty} Q_{n}^{(S)} (x, y, k, a) \frac{w^{n}}{n!} .

By combining the above equation with (7) and (20), we have

\begin{matrix} \frac{a}{2 (k + 2)} \sum_{n = 0}^{\infty} E_{n}^{(S)} (\frac{x}{k}, \frac{y}{k}) k^{n} \frac{w^{n}}{n!} \sum_{n = 0}^{\infty} B_{n} (\frac{k + 1}{k + 2}) {(k + 2)}^{n} \frac{w^{n}}{n!} \\ = & \sum_{n = 0}^{\infty} Q_{n}^{(S)} (x, y, k, a) \frac{w^{n}}{n!} . \end{matrix}

Therefore

\begin{matrix} \frac{a}{2} \sum_{n = 0}^{\infty} \sum_{j = 0}^{n} (\binom{n}{j}) E_{j}^{(S)} (\frac{x}{k}, \frac{y}{k}) B_{n - j} (\frac{k + 1}{k + 2}) k^{j} {(k + 2)}^{n - j - 1} \frac{w^{n}}{n!} \\ = & \sum_{n = 0}^{\infty} Q_{n}^{(S)} (x, y, k, a) \frac{w^{n}}{n!} . \end{matrix}

By comparing the coefficients of

\frac{w^{n}}{n!}

on both sides of the above equation, we get the desired result. □

Theorem 2.

Let

n \in N_{0}

. Then, we have

Q_{n}^{(C)} (x, y, k, a) = \frac{a}{2} \sum_{j = 0}^{n} (\binom{n}{j}) E_{j}^{(C)} (\frac{x}{k}, \frac{y}{k}) B_{n - j} (\frac{k + 1}{k + 2}) k^{j} {(k + 2)}^{n - j - 1} .

Proof.

By using (5), we obtain the following functional equation:

\frac{e^{x w} cos (y w)}{e^{k w} + 1} (\frac{a w}{e^{(k + 2) w} - 1}) e^{(k + 1) w} = \sum_{n = 0}^{\infty} Q_{n}^{(C)} (x, y, k, a) \frac{w^{n}}{n!} .

By combining the above equation with (7) and (19), we have

\begin{matrix} \frac{a}{2 (k + 2)} \sum_{n = 0}^{\infty} E_{n}^{(C)} (\frac{x}{k}, \frac{y}{k}) k^{n} \frac{w^{n}}{n!} \sum_{n = 0}^{\infty} B_{n} (\frac{k + 1}{k + 2}) {(k + 2)}^{n} \frac{w^{n}}{n!} \\ = & \sum_{n = 0}^{\infty} Q_{n}^{(C)} (x, y, k, a) \frac{w^{n}}{n!} . \end{matrix}

Therefore

\begin{matrix} \frac{a}{2} \sum_{n = 0}^{\infty} \sum_{j = 0}^{n} (\binom{n}{j}) E_{j}^{(C)} (\frac{x}{k}, \frac{y}{k}) B_{n - j} (\frac{k + 1}{k + 2}) k^{j} {(k + 2)}^{n - j - 1} \frac{w^{n}}{n!} \\ = & \sum_{n = 0}^{\infty} Q_{n}^{(C)} (x, y, k, a) \frac{w^{n}}{n!} . \end{matrix}

By comparing the coefficients of

\frac{w^{n}}{n!}

on both sides of the above equation, we obtain the desired result. □

Theorem 3.

Let

n \in N

. Then we have

Q_{n}^{(S)} (x, y, k, a) = \sum_{j = 0}^{[\frac{n - 1}{2}]} {(- 1)}^{j} (\binom{n}{2 j + 1}) y^{2 j + 1} Q_{n - 1 - 2 j} (x, k, a) .

Proof.

By using (2) and (4), we obtain

\sum_{n = 0}^{\infty} Q_{n}^{(S)} (x, y, k, a) \frac{w^{n}}{n!} = \sum_{n = 0}^{\infty} {(- 1)}^{n} \frac{{(y w)}^{2 n + 1}}{(2 n + 1)!} \sum_{n = 0}^{\infty} Q_{n} (x, k, a) \frac{w^{n}}{n!} .

Therefore

\sum_{n = 0}^{\infty} Q_{n}^{(S)} (x, y, k, a) \frac{w^{n}}{n!} = \sum_{n = 0}^{\infty} \sum_{j = 0}^{[\frac{n - 1}{2}]} {(- 1)}^{j} (\binom{n}{2 j + 1}) y^{2 j} Q_{n - 1 - 2 j} (x, k, a) \frac{w^{n}}{n!} .

By comparing the coefficients of

\frac{w^{n}}{n!}

on both sides of the above equation, we achieve the desired result. □

Theorem 4.

Let

n \in N_{0}

. Then, we have

Q_{n}^{(C)} (x, y, k, a) = \sum_{j = 0}^{[\frac{n}{2}]} {(- 1)}^{j} (\binom{n}{2 j}) y^{2 j} Q_{n - 2 j} (x, k, a) .

Proof.

By using (2) and (4), we obtain

\sum_{n = 0}^{\infty} Q_{n}^{(C)} (x, y, k, a) \frac{w^{n}}{n!} = \sum_{n = 0}^{\infty} {(- 1)}^{n} \frac{{(y w)}^{2 n}}{(2 n)!} \sum_{n = 0}^{\infty} Q_{n} (x, k, a) \frac{w^{n}}{n!} .

Therefore

\sum_{n = 0}^{\infty} Q_{n}^{(C)} (x, y, k, a) \frac{w^{n}}{n!} = \sum_{n = 0}^{\infty} \sum_{j = 0}^{[\frac{n}{2}]} {(- 1)}^{j} (\binom{n}{2 j}) y^{2 j} Q_{n - 2 j} (x, k, a) \frac{w^{n}}{n!} .

By comparing the coefficients of

\frac{w^{n}}{n!}

on both sides of the above equation, we achieve the desired result. □

Theorem 5.

Let

n \in N_{0}

. Then, we have

Q_{n}^{(S)} (x, y, k, a) = \sum_{j = 0}^{n} (\binom{n}{j}) S_{j} (x, y) Y_{n - j} (k, a) .

Proof.

By using (1), (4), and (16), we obtain

\sum_{n = 0}^{\infty} Q_{n}^{(S)} (x, y, k, a) \frac{w^{n}}{n!} = \sum_{n = 0}^{\infty} S_{n} (x, y) \frac{w^{n}}{n!} \sum_{n = 0}^{\infty} Y_{n} (k, a) \frac{w^{n}}{n!} .

Therefore

\sum_{n = 0}^{\infty} Q_{n}^{(S)} (x, y, k, a) \frac{w^{n}}{n!} = \sum_{n = 0}^{\infty} \sum_{j = 0}^{n} (\binom{n}{j}) S_{j} (x, y) Y_{n - j} (k, a) \frac{w^{n}}{n!} .

Byy omparing the coefficients of

\frac{w^{n}}{n!}

on both sides of the above equation, we achieve the desired result. □

Theorem 6.

Let

n \in N_{0}

. Then, we have

Q_{n}^{(C)} (x, y, k, a) = \sum_{j = 0}^{n} (\binom{n}{j}) C_{j} (x, y) Y_{n - j} (k, a) .

Proof.

By using (1), (4), and (15), we have

\sum_{n = 0}^{\infty} Q_{n}^{(C)} (x, y, k, a) \frac{w^{n}}{n!} = \sum_{n = 0}^{\infty} C_{n} (x, y) \frac{w^{n}}{n!} \sum_{n = 0}^{\infty} Y_{n} (k, a) \frac{w^{n}}{n!} .

Therefore

\sum_{n = 0}^{\infty} Q_{n}^{(C)} (x, y, k, a) \frac{w^{n}}{n!} = \sum_{n = 0}^{\infty} \sum_{j = 0}^{n} (\binom{n}{j}) C_{j} (x, y) Y_{n - j} (k, a) \frac{w^{n}}{n!} .

By comparing the coefficients of

\frac{w^{n}}{n!}

on both sides of the above equation, we achieve the desired result. □

Theorem 7.

Let

n \in N_{0}

. Then, we have

Q_{n} (x + i y, k, a) = Q_{n}^{(C)} (x, y, k, a) + i Q_{n}^{(S)} (x, y, k, a) .

Proof.

By using (4) and (5), we have

\sum_{n = 0}^{\infty} (Q_{n}^{(C)} (x, y, k, a) + i Q_{n}^{(S)} (x, y, k, a)) \frac{w^{n}}{n!} = \frac{e^{x w + i y w} a w}{e^{(k + 1) w} - e^{- (k + 1) w} + e^{w} - e^{- w}} .

By using the the above equation and the Euler’s formula, we obtain

\sum_{n = 0}^{\infty} (Q_{n}^{(C)} (x, y, k, a) + i Q_{n}^{(S)} (x, y, k, a)) \frac{w^{n}}{n!} = \sum_{n = 0}^{\infty} Q_{n} (x + i y, k, a) \frac{w^{n}}{n!} .

By comparing the coefficients of

\frac{w^{n}}{n!}

on both sides of the above equation, we get the desired result. □

Theorem 8.

Let

n \in N_{0}

. Then, we have

\begin{matrix} \sum_{v = 0}^{n} (\binom{n}{v}) \sum_{j = 0}^{v} (\binom{v}{j}) Q_{j}^{(C)} (x, y, k, 2) Q_{v - j}^{(S)} (x, y, k, 2) k^{n - v} E_{n - v}^{(- 2)} \\ = & \frac{{(k + 2)}^{n - 2}}{2} B_{n}^{(C, 2)} (\frac{2 (k + 1 + x)}{k + 2}, \frac{2 y}{k + 2}), \end{matrix}

where

B_{n}^{(C, 2)} (\frac{2 (k + 1 + x)}{k + 2}, \frac{2 y}{k + 2}) = \sum_{v = 0}^{n} (\binom{n}{v}) B_{v}^{(2)} S_{n - v} (\frac{2 (k + 1 + x)}{k + 2}, \frac{2 y}{k + 2}) .

Proof.

By using (4) and (5), we have

\begin{matrix} \sum_{n = 0}^{\infty} Q_{n}^{(C)} (x, y, k, 2) \frac{w^{n}}{n!} \sum_{n = 0}^{\infty} Q_{n}^{(S)} (x, y, k, 2) \frac{w^{n}}{n!} \\ = & \frac{1}{2} (\frac{w^{2} e^{2 (k + 1 + x) w} sin (2 y w)}{{(e^{(k + 2) w} - 1)}^{2}}) {(\frac{2}{e^{k w} + 1})}^{2} . \end{matrix}

By combining the above equation with (8) and (17), we obtain

\begin{matrix} \sum_{n = 0}^{\infty} \sum_{j = 0}^{n} (\binom{n}{j}) Q_{j}^{(C)} (x, y, k, 2) Q_{n - j}^{(S)} (x, y, k, 2) \frac{w^{n}}{n!} \sum_{n = 0}^{\infty} E_{n}^{(- 2)} \frac{{(k w)}^{n}}{n!} \\ = & \frac{1}{2 {(k + 2)}^{2}} \sum_{n = 0}^{\infty} B_{n}^{(C, 2)} (\frac{2 (k + 1 + x)}{k + 2}, \frac{2 y}{k + 2}) {(k + 2)}^{n} \frac{w^{n}}{n!} . \end{matrix}

Therefore

\begin{matrix} \sum_{n = 0}^{\infty} \sum_{v = 0}^{n} (\binom{n}{v}) \sum_{j = 0}^{v} (\binom{v}{j}) Q_{j}^{(C)} (x, y, k, 2) Q_{v - j}^{(S)} (x, y, k, 2) k^{n - v} E_{n - v}^{(- 2)} \frac{w^{n}}{n!} \\ = & \frac{1}{2} \sum_{n = 0}^{\infty} B_{n}^{(C, 2)} (\frac{2 (k + 1 + x)}{k + 2}, \frac{2 y}{k + 2}) {(k + 2)}^{n - 2} \frac{w^{n}}{n!} . \end{matrix}

By comparing the coefficients of

\frac{w^{n}}{n!}

on both sides of the above equation, we achieve the desired result. □

In the following we state interesting identities related to the Stirling numbers, the polynomials

C_{m} (x, y)

and

S_{m} (x, y)

By combining (12) with (15) and (16), we get the following functional equations:

cos (y w) \sum_{n = 0}^{\infty} {(x)}_{n} F_{S} (w, n) = K_{C} (w, x, y)

(23)

and

sin (y w) \sum_{n = 0}^{\infty} {(x)}_{n} F_{S} (w, n) = K_{S} (w, x, y) .

(24)

Theorem 9.

Let

m \in N_{0}

. Then we have

C_{m} (x, y) = \sum_{j = 0}^{[\frac{m}{2}]} {(- 1)}^{j} y^{2 j} (\binom{m}{2 j}) \sum_{n = 0}^{m - 2 j} {(x)}_{n} S_{2} (m - 2 j, n) .

(25)

Proof.

By using (23), we obtain

\sum_{m = 0}^{\infty} {(- 1)}^{m} \frac{{(y w)}^{2 m}}{(2 m)!} \sum_{m = 0}^{\infty} \sum_{n = 0}^{m} {(x)}_{n} S_{2} (m, n) \frac{w^{m}}{m!} = \sum_{m = 0}^{\infty} C_{m} (x, y) \frac{w^{m}}{m!} .

Therefore

\sum_{m = 0}^{\infty} C_{m} (x, y) \frac{w^{m}}{m!} = \sum_{m = 0}^{\infty} \sum_{j = 0}^{[\frac{m}{2}]} {(- 1)}^{j} y^{2 j} (\binom{m}{2 j}) \sum_{n = 0}^{m - 2 j} {(x)}_{n} S_{2} (m - 2 j, n) \frac{w^{m}}{m!} .

By comparing the coefficients of

\frac{w^{m}}{m!}

on both sides of the above equation, we obtain the desired result. □

By combining (25) with (14), we arrive at the following theorem:

Theorem 10.

Let

m \in N_{0}

. Then, we have

\begin{matrix} C_{m} (x, y) & = & \sum_{j = 0}^{[\frac{m}{2}]} {(- 1)}^{j} y^{2 j} (\binom{m}{2 j}) \sum_{n = 0}^{m - 2 j} \frac{{(x)}_{n} 2^{n - m + 2 j}}{n!} \\ \times \sum_{v = 0}^{m - 2 j} \sum_{d = 0}^{n} {(- 1)}^{n - d} (\binom{m - 2 j}{v}) (\binom{n}{d}) d^{v} E_{m - 2 j - v}^{(- n)} . \end{matrix}

By combining (25) with (22), we arrive at the following theorem:

Theorem 11.

Let

m \in N_{0}

. Then, we have

C_{m} (x, y) = \sum_{j = 0}^{[\frac{m}{2}]} {(- 1)}^{j} \frac{{(2 y)}^{2 j} β_{2 j} (m)}{(2 j)!} \sum_{n = 0}^{m - 2 j} {(x)}_{n} S_{2} (m - 2 j, n) .

Theorem 12.

Let

m \in N

. Then, we have

S_{m} (x, y) = \sum_{j = 0}^{[\frac{m - 1}{2}]} {(- 1)}^{j} (\binom{m}{2 j + 1}) y^{2 j + 1} \sum_{n = 0}^{m - 1 - 2 j} {(x)}_{n} S_{2} (m - 1 - 2 j, n) .

(26)

Proof.

By using (24), we obtain

\sum_{m = 0}^{\infty} S_{m} (x, y) \frac{w^{m}}{m!} = \sum_{n = 0}^{\infty} {(- 1)}^{n} \frac{{(y w)}^{2 n + 1}}{(2 n + 1)!} \sum_{m = 0}^{\infty} \sum_{n = 0}^{m} {(x)}_{n} S_{2} (m, n) \frac{w^{m}}{m!} .

Therefore

\begin{matrix} \sum_{m = 0}^{\infty} S_{m} (x, y) \frac{w^{m}}{m!} & = & \sum_{m = 0}^{\infty} m \sum_{j = 0}^{[\frac{m - 1}{2}]} {(- 1)}^{j} (\binom{m - 1}{2 j}) \frac{y^{2 j + 1}}{2 j + 1} \\ \times \sum_{n = 0}^{m - 1 - 2 j} {(x)}_{n} S_{2} (m - 1 - 2 j, n) \frac{w^{m}}{m!} . \end{matrix}

By comparing the coefficients of

\frac{w^{m}}{m!}

on both sides of the above equation, we obtain the desired result. □

Remark 1.

By combining (13)with (15)and (16), we also arrive at Formulas (25)and (26). For these and similar formulas, the above formula may be used.

3. Questions

In [33], Kim, and Simsek gave interpolation functions involving the Hurwitz zeta function and the alternating Hurwitz zeta function for the numbers

Y_{n} (k, a)

How can we define interpolation functions for the following polynomials:

Q_{n} (x, k, a), Q_{n}^{(C)} (x, y, k, a), a n d Q_{n}^{(S)} (x, y, k, a) ?

4. Conclusions

Applications of generating functions are used in a remarkably wide range of areas, and we used them to define new classes of parametric kinds of special polynomials. By using the method of generating functions and Euler’s formula, we investigated properties of these new parametric kinds of special polynomials. We also provide new identities and relations involving these classes of special polynomials, the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the cosine-Bernoulli polynomials, the sine-Bernoulli polynomials, the cosine-Euler polynomials, the sine-Euler polynomials, and known special polynomials.

In the near future, with the help of the results given in this article, solutions of the open questions put forward will be investigated.

In general, these results have the potential to be used many branches of mathematics, probability, statistics, mathematical physics, and engineering.

Author Contributions

A.B. and Y.S. contributed equally to the content of the paper. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Acknowledgments

The second-named author was supported by the Scientific Research Project Administration of the University of Akdeniz.

Conflicts of Interest

The authors declare no conflict of interest.

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Bayad, A.; Simsek, Y. On Generating Functions for Parametrically Generalized Polynomials Involving Combinatorial, Bernoulli and Euler Polynomials and Numbers.Symmetry2022,14, 654. https://doi.org/10.3390/sym14040654

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Bayad A, Simsek Y. On Generating Functions for Parametrically Generalized Polynomials Involving Combinatorial, Bernoulli and Euler Polynomials and Numbers.Symmetry. 2022; 14(4):654. https://doi.org/10.3390/sym14040654

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Bayad, Abdelmejid, and Yilmaz Simsek. 2022. "On Generating Functions for Parametrically Generalized Polynomials Involving Combinatorial, Bernoulli and Euler Polynomials and Numbers"Symmetry 14, no. 4: 654. https://doi.org/10.3390/sym14040654

APA Style

Bayad, A., & Simsek, Y. (2022). On Generating Functions for Parametrically Generalized Polynomials Involving Combinatorial, Bernoulli and Euler Polynomials and Numbers.Symmetry,14(4), 654. https://doi.org/10.3390/sym14040654

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On Generating Functions for Parametrically Generalized Polynomials Involving Combinatorial, Bernoulli and Euler Polynomials and Numbers

Abstract

1. Motivation and Preliminaries

1.1. Motivation

1.2. Preliminaries

2. Generating Functions for New Classes of Parametric Kinds of Special Polynomials

3. Questions

4. Conclusions

Author Contributions

Funding

Acknowledgments

Conflicts of Interest

References

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