Hermite-based unified Apostol-Bernoulli, Euler and Genocchi polynomials

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Abstract

In this paper, we introduce a unified family of Hermite-based Apostol-Bernoulli, Euler and Genocchi polynomials. We obtain some symmetry identities between these polynomials and the generalized sum of integer powers. We give explicit closed-form formulae for this unified family. Furthermore, we prove a finite series relation between this unification and 3d-Hermite polynomials.

MSC:11B68, 33C05.

1 Introduction

Recently, Khan et al.  introduced the Hermite-based Appell polynomials via the generating function

$G(x,y,z;t)=A(t)exp(Mt),$

where

$M=x+2y ∂ ∂ x +3z ∂ 2 ∂ x 2$

is the multiplicative operator of the 3-variable Hermite polynomials, which are defined by

$exp ( x t + y t 2 + z t 3 ) = ∑ n = 0 ∞ H n ( 3 ) (x,y,z) t n n !$
(1.1)

and

$A(t)= ∑ n = 0 ∞ a n t n , a 0 ≠0.$

By using the Berry decoupling identity,

$e A + B = e m 2 / 12 e ( ( − m 2 ) A 1 / 2 + A ) e B ,[A,B]=m A 1 / 2$

they obtained the generating function of the Hermite-based Appell polynomials $A n H (x,y,z)$ as

$G(x,y,z;t)=A(t)exp ( x t + y t 2 + z t 3 ) = ∑ n = 0 ∞ H A n (x,y,z) t n n ! .$

Letting $A(t)= t e t − 1$, they defined Hermite-Bernoulli polynomials $B n H (x,y,z)$ by

$t e t − 1 exp ( x t + y t 2 + z t 3 ) = ∑ n = 0 ∞ H B n (x,y,z) t n n ! ,|t|<2π.$

For $A(t)= 2 e t + 1$, they defined Hermite-Euler polynomials $E n H (x,y,z)$ by

$2 e t + 1 exp ( x t + y t 2 + z t 3 ) = ∑ n = 0 ∞ H E n (x,y,z) t n n ! ,|t|<π$

and for $A(t)= 2 t e t + 1$, they defined Hermite-Genocchi polynomials $G n H (x,y,z)$ by

$2 t e t + 1 exp ( x t + y t 2 + z t 3 ) = ∑ n = 0 ∞ H G n (x,y,z) t n n ! ,|t|<π.$

Recently, the author considered the following unification of the Apostol-Bernoulli, Euler and Genocchi polynomials

$f a , b ( α ) ( x ; t ; k , β ) : = ( 2 1 − k t k β b e t − a b ) α e x t = ∑ n = 0 ∞ P n , β ( α ) ( x ; k , a , b ) t n n ! ( k ∈ N 0 ; a , b ∈ R ∖ { 0 } ; α , β ∈ C )$

and obtained the explicit representation of this unified family, in terms of Gaussian hypergeometric function. Some symmetry identities and multiplication formula are also given in . Note that the family of polynomials $P n , β ( 1 ) (x,y,z;k,a,b)$ was investigated in .

We organize the paper as follows.

In Section 2, we introduce the unification of the Hermite-based generalized Apostol-Bernoulli, Euler and Genocchi polynomials $P n , β ( α ) H (x,y,z;k,a,b)$ and give summation formulas for this unification. In Section 3, we obtain some symmetry identities for these polynomials. In Section 4, we give explicit closed-form formulae for this unified family. Furthermore, we prove a finite series relation between this unification and 3d-Hermite polynomials.

2 Hermite-based generalized Apostol-Bernoulli, Euler and Genocchi polynomials

In this paper, we consider the following general class of polynomials:

$f a , b ( α ) ( x , y , z ; t ; k , β ) : = ( 2 1 − k t k β b e t − a b ) α e x t + y t 2 + z t 3 = ∑ n = 0 ∞ H P n , β ( α ) ( x , y , z ; k , a , b ) t n n ! ( k ∈ N 0 ; a , b ∈ R ∖ { 0 } ; α , β ∈ C ) .$
(2.1)

For the existence of the expansion, we need

1. (i)

$|t|<2π$ when $α∈C$, $k=1$ and $( β a ) b =1$; $|t|<2π$ when $α∈ N 0$, $k=2,3,…$ and $( β a ) b =1$; $|t|<|blog( β a )|$ when $α∈ N 0$, $k∈N$ and $( β a ) b ≠1$ (or $≠−1$); $x,y,z∈R$, $β∈C$, $a,b∈C/{0}$; $1 α :=1$;

2. (ii)

$|t|<π$ when $( β a ) b =−1$; $|t|<|blog( β a )|$ when $( β a ) b ≠−1$; $x,y,z∈R$, $k=0$, $α,β∈C$, $a,b∈C/{0}$; $1 α :=1$;

3. (iii)

$|t|<π$ when $α∈ N 0$ and $( β a ) b =−1$; $x,y,z∈R$, $k∈N$, $β∈C$, $a,b∈C/{0}$; $1 α :=1$,

where $w=|w| e i θ$, $−π≤θ<π$ and $log(w)=log(|w|)+iθ$.

For $k=a=b=1$ and $β=λ$ in (2.1), we define the following.

Definition 2.1 Let $α∈ N 0$, λ be an arbitrary (real or complex) parameter and $x,y,z∈R$. The Hermite-based generalized Apostol-Bernoulli polynomials are defined by

It is clear that

$P n , λ ( α ) H (x,y,z;1,1,1) = H B n ( α ) (x,y,z;λ).$

Some special cases of the Hermite-based generalized Apostol-Bernoulli polynomials (some of which are definition) are listed below:

• $B n ( 1 ) H (x,y,z;λ): = H B n (x,y,z;λ)$ is called Hermite-based Apostol-Bernoulli polynomials.

• $B n H (x,y,z;1) = H B n (x,y,z)$ is the Hermite-Bernoulli polynomials.

• $B n H (x,0,0;λ):= B n (x;λ)$ is the Apostol-Bernoulli polynomials (see ). When $λ=1$, we have the classical Bernoulli polynomials.

• $B n (0;λ):= B n (λ)$ are the Apostol-Bernoulli numbers. $λ=1$ gives the classical Bernoulli numbers.

Setting $k+1=−a=b=1$ and $β=λ$ in (2.1), we get the following.

Definition 2.2 Let α and λ ($≠−1$) be an arbitrary (real or complex) parameter and $x,y,z∈R$. The Hermite-based generalized Apostol-Euler polynomials are defined by

Obviously, we have

$P n , λ ( α ) H (x,y,z;0,−1,1) = H E n ( α ) (x,y,z;λ).$

Some special cases of the Hermite-based generalized Apostol-Euler polynomials (some of which are definition) are listed below:

• $E n ( 1 ) H (x,y,z;λ): = H E n (x,y,z;λ)$ is called Hermite-based Apostol-Euler polynomials.

• $E n H (x,y,z;1) = H E n (x,y,z)$ is the Hermite-Euler polynomials.

• $E n H (x,0,0;λ):= E n (x;λ)$ is the Apostol-Euler polynomials (see ). For $λ=1$, we have the classical Euler polynomials.

• $2 n E n ( 1 2 ;λ):= E n (λ)$ are the Apostol-Euler numbers. The case $λ=1$ gives the classical Euler numbers.

Choosing $k=−2a=b=1$ and $2β=λ$ in (2.1), we define the following.

Definition 2.3 Let α and λ ($≠−1$) be an arbitrary (real or complex) parameter and $x,y,z∈R$. The Hermite-based generalized Apostol-Genocchi polynomials are defined by

It is easily seen that

$P n , λ 2 ( α ) H ( x , y , z ; 1 , − 1 2 , 1 ) = H G n α (x,y,z;λ).$

Some special cases of the Hermite-based generalized Apostol-Genocchi polynomials (some of which are definition) are listed below:

• $G n ( 1 ) H (x,y,z;λ): = H G n (x,y,z;λ)$ is called Hermite-based Apostol-Genocchi polynomials.

• $G n H (x,y,z;1) = H G n (x,y,z)$ is the Hermite-Genocchi polynomials.

• $G n H (x,0,0;λ):= G n (x;λ)$ is the Apostol-Genocchi polynomials (see [9, 10]). When $λ=1$, we have the classical Genocchi polynomials.

• $G n (0;λ):= G n (λ)$ are the Apostol-Genocchi numbers. $λ=1$ gives the classical Genocchi numbers.

Finally we define the unified Hermite-based Apostol polynomials by

$f a , b ( 1 ) ( x ; t ; k , β ) : = 2 1 − k t k β b e t − a b e x t + y t 2 + z t 3 = ∑ n = 0 ∞ H P n , β ( x , y , z ; k , a , b ) t n n ! ( k ∈ N 0 ; a , b ∈ R ∖ { 0 } ; β ∈ C ) .$

Thus it is clear that $P n , β H (x,y,z;k,a,b) = H P n , β ( 1 ) (x,y,z;k,a,b)$ and that we have the following observations at once:

• $P n , λ H (x,y,z;1,1,1) = H B n (x,y,z;λ)$ are the Hermite-based Apostol-Bernoulli polynomials.

• $P n , λ H (x,y,z;0,−1,1) = H E(x,y,z;λ)$ are the Hermite-based Apostol-Euler polynomials.

• $P n , λ 2 H (x,y,z;1, − 1 2 ,1) = H G n (x,y,z;λ)$ are the Hermite-based Apostol-Genocchi polynomials.

For the other generalization, we refer  and . Now we give some relations between the above mentioned Apostol polynomials.

Using (2.1), we get the following identity at once.

Theorem 2.1 Let $α,k∈ N 0$; $a,b∈R∖{0}$; $β∈C$ be such that the conditions (i)-(iii) are satisfied. Then, the following relation

$∑ r = 0 n ( n r ) H P n − r , β ( α ) ( x , y , z ; k , a , b ) H P r , β ( α ) (u,v,w;k,a,b) = H P n , β ( α ) (x+u,y+v,z+w;k,a,b)$

holds true.

Corollary 2.2 For each $n∈N$, the following relation

$∑ k = 0 n ( n k ) H B n − k ( α ) ( x , y , z ; λ ) H B k ( β ) (u,v,w;λ) = H B n ( α + β ) (x+u,y+v,z+w;λ)$

holds true for the Hermite-based generalized Apostol-Bernoulli polynomials.

Corollary 2.3 For each $n∈N$, the following relation

$∑ k = 0 n ( n k ) H E n − k ( α ) ( x , y , z ; λ ) H E k ( β ) (u,v,w;λ) = H E n ( α + β ) (x+u,y+v,z+w;λ)$

holds true for the Hermite-based generalized Apostol-Euler polynomials.

Corollary 2.4 For each $n∈N$, the following relation

$∑ k = 0 n ( n k ) H G n − k ( α ) ( x , y , z ; λ ) H G k ( β ) (u,v,w;λ) = H G n ( α + β ) (x+u,y+v,z+w;λ)$

holds true for the Hermite-based generalized Apostol-Genocchi polynomials.

Theorem 2.5 For each $n∈N$, the following relation

$∑ k = 0 n ( n k ) H B n − k ( α ) ( x , y , z ; λ ) H E k ( α ) (u,v,w;λ)= 2 H n B n ( α ) ( x + u 2 , y + v 4 , z + w 8 ; λ 2 )$

holds true between the Hermite-based generalized Apostol-Bernoulli and Euler polynomials.

Proof By direct calculations, we have

$∑ n = 0 ∞ H B n ( α ) ( x + u 2 , y + v 4 , z + w 8 ; λ 2 ) ( 2 t ) n n ! = ( 2 t λ 2 e 2 t − 1 ) α exp [ ( x + u 2 ) 2 t + ( y + v 4 ) ( 2 t ) 2 + ( z + w 8 ) ( 2 t ) 3 ] = ( t λ e t − 1 ) α exp ( x t + y t 2 + z t 3 ) ( 2 λ e t + 1 ) α exp ( u t + v t 2 + w t 3 ) = ∑ n = 0 ∞ H B n ( α ) ( x , y , z ; λ ) t n n ! ∑ k = 0 ∞ H E k ( α ) ( u , v , w ; λ ) t k k ! = ∑ n = 0 ∞ ∑ k = 0 n ( n k ) H B n − k ( α ) ( x , y , z ; λ ) H E k ( α ) ( u , v , w ; λ ) t n n ! .$

Comparing the coefficients of $t n n !$ on both sides, we get the result. □

3 Symmetry identities for the unified family

For each $k∈ N 0$, the sum $S k (n)= ∑ i = 0 n i k$ is known as the power sum and we have the following generating relation:

$∑ k = 0 ∞ S k (n) t k k ! =1+ e t + e 2 t +⋯+ e n t = e ( n + 1 ) t − 1 e t − 1 .$

For an arbitrary real or complex λ, the generalized sum of integer powers $S k (n,λ)$ is defined, in , via the following generating relation:

$∑ k = 0 ∞ S k (n,λ) t k k ! = λ e ( n + 1 ) t − 1 λ e t − 1 .$

It clear that $S k (n,1)= S k (n)$.

For each $k∈ N 0$, the sum $M k (n)= ∑ i = 0 n ( − 1 ) k i k$ is known as the sum of alternative integer powers. The following generating relation is straightforward:

$∑ k = 0 ∞ M k (n) t k k ! =1− e t + e 2 t −⋯+ ( − 1 ) n e n t = 1 − ( − e t ) ( n + 1 ) e t + 1 .$

For an arbitrary real or complex λ, the generalized sum of alternative integer powers $M k (n,λ)$ is defined, in , by

$∑ k = 0 ∞ M k (n,λ) t k k ! = 1 − λ ( − e t ) ( n + 1 ) λ e t + 1 .$

Clearly $M k (n,1)= M k (n)$. On the other hand, if n is even, then

$S k (n,−λ)= M k (n,λ).$
(3.1)

We start by obtaining certain symmetry identities, which includes the results given in  and , when $y=z=0$.

Theorem 3.1 Let $c,d,m∈N$, $n∈ N 0$ be such that the conditions (i)-(iii) are satisfied with t replaced by ct and dt. Then we have the following symmetry identity:

$∑ r = 0 n ( n r ) c n − r d r + k H P n − r , β ( m ) ( d x , d 2 y , d 3 z ; k , a , b ) × ∑ l = 0 r ( r l ) S l ( c − 1 ; ( β a ) b ) H P r − l , β ( m − 1 ) ( c X , c 2 Y , c 3 Z ; k , a , b ) = ∑ r = 0 n ( n r ) d n − r c r + k H P n − r , β ( m ) ( c x , c 2 y , c 3 z ; k , a , b ) × ∑ l = 0 r ( r l ) S l ( d − 1 ; ( β a ) b ) H P r − l , β ( m − 1 ) ( d X , d 2 Y , d 3 Z ; k , a , b ) .$

Proof Let

$G(t):= 2 ( 1 − k ) ( 2 m − 1 ) t 2 k m − k e c d x t + y ( c d t ) 2 + z ( c d t ) 3 ( β b e c d t − a b ) e c d X t + Y ( c d t ) 2 + Z ( c d t ) 3 ( β b e c t − a b ) m ( β b e d t − a b ) m .$

Expanding $G(t)$ into a series, we get

$G ( t ) = 1 c k m d k ( m − 1 ) ( 2 1 − k c k t k β b e c t − a b ) m e c d x t + y ( c d t ) 2 + z ( c d t ) 3 ( β b e c d t − a b β b e d t − a b ) × ( 2 1 − k d k t k β b e d t − a b ) m − 1 e c d X t + Y ( c d t ) 2 + Z ( c d t ) 3 = 1 c k m d k ( m − 1 ) [ ∑ n = 0 ∞ H P n , β ( m ) ( d x , d 2 y , d 3 z ; k , a , b ) ( c t ) n n ! ] [ ∑ l = 0 ∞ S l ( c − 1 ; ( β a ) b ) ( d t ) l l ! ] × [ ∑ r = 0 ∞ H P r , β ( m − 1 ) ( c X , c 2 Y , c 3 Z ; k , a , b ) ( d t ) r r ! ] .$

Now, using Corollary 2 in [, p.890], we get

$G ( t ) = 1 c k m d k m ∑ n = 0 ∞ [ ∑ r = 0 n ( n r ) c n − r d r + k H P n − r , β ( m ) ( d x , d 2 y , d 3 z ; k , a , b ) × ∑ l = 0 r ( r l ) S l ( c − 1 ; ( β a ) b ) H P r − l , β ( m − 1 ) ( c X , c 2 Y , c 3 Z ; k , a , b ) ] t n n ! .$
(3.2)

In a similar manner,

$G ( t ) = 1 d k m c k ( m − 1 ) ( 2 1 − k d k t k β b e c t − a b ) m e c d x t + y ( c d t ) 2 + z ( c d t ) 3 ( β b e c d t − a b β b e d t − a b ) × ( 2 1 − k c k t k β b e d t − a b ) m − 1 e c d X t + Y ( c d t ) 2 + Z ( c d t ) 3 = 1 c k m d k m ∑ n = 0 ∞ [ ∑ r = 0 n ( n r ) d n − r c r + k H P n − r , β ( m ) ( c x , c 2 y , c 3 z ; k , a , b ) × ∑ l = 0 r ( r l ) S l ( d − 1 ; ( β a ) b ) H P r − l , β ( m − 1 ) ( d X , d 2 Y , d 3 Z ; k , a , b ) ] t n n ! .$
(3.3)

From (3.2) and (3.3), we get the result. □

For $k=a=b=1$ and $β=λ$ we get the following corollary at once.

Corollary 3.2 For all $c,d,m∈N$, $n∈ N 0$, $λ∈C$, we have the following symmetry identity for the Hermite based generalized Apostol-Bernoulli polynomials:

$∑ r = 0 n ( n r ) c n − r d r + 1 H B n − r ( m ) ( d x , d 2 y , d 3 z , λ ) × ∑ l = 0 r ( r l ) S l ( c − 1 ; λ ) H B r − l ( m − 1 ) ( c X , c 2 Y , c 3 Z , λ ) = ∑ r = 0 n ( n r ) d n − r c r + 1 H B n − r ( m ) ( c x , c 2 y , c 3 z , λ ) × ∑ l = 0 r ( r l ) S l ( d − 1 ; λ ) H B r − l ( m − 1 ) ( d X , d 2 Y , d 3 Z , λ ) .$

For $k+1=−a=b=1$ and $β=λ$ we get, by considering (3.1) that

Corollary 3.3 For all $m∈N$, $n∈ N 0$, $λ∈C$, we have for each pair of positive even integers c and d, or for each pair of positive odd integers c and d,

$∑ r = 0 n ( n r ) c n − r d r + 1 H E n − r ( m ) ( d x , d 2 y , d 3 z , λ ) × ∑ l = 0 r ( r l ) M l ( c − 1 ; λ ) H E r − l ( m − 1 ) ( c X , c 2 Y , c 3 Z , λ ) = ∑ r = 0 n ( n r ) d n − r c r + 1 H E n − r ( m ) ( c x , c 2 y , c 3 z , λ ) × ∑ l = 0 r ( r l ) M l ( d − 1 ; λ ) H E r − l ( m − 1 ) ( d X , d 2 Y , d 3 Z , λ ) .$

Letting $k=−2a=b=1$ and $2β=λ$ and taking into account (3.1) that we have the following.

Corollary 3.4 For all $m∈N$, $n∈ N 0$, $λ∈C$, we have for each pair of positive even integers c and d, or for each pair of positive odd integers c and d, that

$∑ r = 0 n ( n r ) c n − r d r + 1 H G n − r ( m ) ( d x , d 2 y , d 3 z , λ ) × ∑ l = 0 r ( r l ) M l ( c − 1 ; λ ) H G r − l ( m − 1 ) ( c X , c 2 Y , c 3 Z , λ ) = ∑ r = 0 n ( n r ) d n − r c r + 1 H G n − r ( m ) ( c x , c 2 y , c 3 z , λ ) × ∑ l = 0 r ( r l ) M l ( d − 1 ; λ ) H G r − l ( m − 1 ) ( d X , d 2 Y , d 3 Z , λ ) .$

4 Closed-form formulae for Hermite-based generalized Apostol polynomials

In this section, taking into account the relations

$f a , b ( α ) ( x , y , z ; t ; k , β ) : = ( 2 1 − k t k β b e t − a b ) α e x t + y t 2 + z t 3 = ∑ n = 0 ∞ H P n , β ( α ) ( x , y , z ; k , a , b ) t n n ! , f a , b ( 1 ) ( x , y , z ; t ; k , β ) : = ( 2 1 − k t k β b e t − a b ) e x t + y t 2 + z t 3 = ∑ n = 0 ∞ H P n , β ( x , y , z ; k , a , b ) t n n ! ,$

we observe the following fact:

$[ f a , b ( 1 ) ( x α , y α , z α ; t ; k , β ) ] α = f a , b ( α ) (x,y,z;t;k,β).$
(4.1)

Using (4.1), we start by proving the following closed form summation formula:

Theorem 4.1 Let the conditions (i)-(iii) be satisfied. The following summation formula:

$∑ l = 0 n ( n l ) [ H P n − l + 1 , β ( α ) ( x , y , z ; k , a , b ) H P l , β ( x α , y α , z α ; k , a , b ) − α H P n − l , β ( α ) ( x , y , z ; k , a , b ) H P l + 1 , β ( x α , y α , z α ; k , a , b ) ] = 0$

holds true.

Proof Taking logarithms on both sides of (4.1) and then differentiating with respect to t, we get

$∂ f a , b ( α ) ( x , y , z ; t ; k , β ) ∂ t f a , b ( 1 ) ( x α , y α , z α ; t ; k , β ) = α f a , b ( α ) ( x , y , z ; t ; k , β ) ∂ f a , b ( 1 ) ( x α , y α , z α ; t ; k , β ) ∂ t .$

Inserting the corresponding generating relations, we obtain

$∑ n = 1 ∞ n H P n , β ( α ) ( x , y , z ; k , a , b ) t n − 1 n ! ∑ l = 0 ∞ H P l , β ( x α , y α , z α ; k , a , b ) t l l ! = α ∑ n = 0 ∞ H P n , β ( α ) ( x , y , z ; k , a , b ) t n n ! ∑ l = 0 ∞ l H P l , β ( x α , y α , z α ; k , a , b ) t l − 1 l ! ,$

and hence

$∑ n = 0 ∞ H P n + 1 , β ( α ) ( x , y , z ; k , a , b ) t n n ! ∑ l = 0 ∞ H P l , β ( x α , y α , z α ; k , a , b ) t l l ! = α ∑ n = 0 ∞ H P n , β ( α ) ( x , y , z ; k , a , b ) t n n ! ∑ l = 0 ∞ H P l + 1 , β ( x α , y α , z α ; k , a , b ) t l l ! .$

Using the fact that (see [, p.101, Lemma 3])

$∑ n = 0 ∞ ∑ l = 0 ∞ A(n,l)= ∑ n = 0 ∞ ∑ l = 0 n A(n−l,l),$
(4.2)

we get

$∑ n = 0 ∞ [ ∑ l = 0 n ( n l ) H P n − l + 1 , β ( α ) ( x , y , z ; k , a , b ) H P l , β ( x α , y α , z α ; k , a , b ) ] t n n ! = α ∑ n = 0 ∞ [ ∑ l = 0 n ( n l ) H P n − l , β ( α ) ( x , y , z ; k , a , b ) H P l + 1 , β ( x α , y α , z α ; k , a , b ) ] t n n ! .$

Whence the result. □

Corollary 4.2 Let $k=a=b=1$ and $β=λ$. For all $m∈N$, $n∈ N 0$, $λ∈C$, we have the following closed form summation formula for the generalized Apostol-Bernoulli polynomials:

$∑ k = 0 n ( n k ) [ B n − k + 1 ( α ) H ( x , y , z ; λ ) H B k ( x α , y α , z α ; λ ) − α H B n − k ( α ) ( x , y , z ; λ ) B k + 1 ( x α , y α , z α ; λ ) ] = 0 .$

Corollary 4.3 Let $k+1=−a=b=1$ and $β=λ$. For all $m∈N$, $n∈ N 0$, $λ∈C$, we have the following closed form summation formula for the generalized Apostol-Euler polynomials:

$∑ k = 0 n ( n k ) [ E n − k + 1 ( α ) H ( x , y , z ; λ ) H E k ( x α , y α , z α ; λ ) − α H E n − k ( α ) ( x , y , z ; λ ) E k + 1 ( x α , y α , z α ; λ ) ] = 0 .$

Corollary 4.4 Let $k=−2a=b=1$ and $2β=λ$. For all $m∈N$, $n∈ N 0$, $λ∈C$, we have the following closed form summation formula for the generalized Apostol-Genocchi polynomials:

$∑ k = 0 n ( n k ) [ G n − k + 1 ( α ) H ( x , y , z ; λ ) H G k ( x α , y α , z α ; λ ) − α H G n − k ( α ) ( x , y , z ; λ ) G k + 1 ( x α , y α , z α ; λ ) ] = 0 .$

Theorem 4.5 Let the conditions (i)-(iii) be satisfied. Then we have the following relation between Hermite based Apostol polynomials and 3d-Hermite polynomials:

$P n + m , β ( α ) H ( X , Y , Z ; k , a , b ) = ∑ r , l = 0 n , m ( n r ) ( m l ) H r + l ( 3 ) ( X − x , Y − y , Z − z ) H P n + m − r − l ( α ) ( x , y , z ; k , a , b ) .$

Proof From (2.1), we can write that

$( 2 1 − k ( t + w ) k β b e t + w − a b ) α e x ( t + w ) + y ( t + w ) 2 + z ( t + w ) 3 = ∑ n = 0 ∞ H P n , β ( α ) ( x , y , z ; k , a , b ) ( t + w ) n n ! = ∑ n , m = 0 ∞ H P n + m , β ( α ) ( x , y , z ; k , a , b ) t n n ! w m m ! .$
(4.3)

Therefore, we get

$( 2 1 − k ( t + w ) k β b e t + w − a b ) α = e − x ( t + w ) − y ( t + w ) 2 − z ( t + w ) 3 ∑ n , m = 0 ∞ H P n + m , β ( α ) (x,y,z;k,a,b) t n n ! w m m ! .$

Multiplying both sides by $e X ( t + w ) + Y ( t + w ) 2 + Z ( t + w ) 3$, we have

$( 2 1 − k ( t + w ) k β b e t + w − a b ) α e X ( t + w ) + Y ( t + w ) 2 + Z ( t + w ) 3 = e ( X − x ) ( t + w ) + ( Y − y ) ( t + w ) 2 + ( Z − z ) ( t + w ) 3 ∑ n , m = 0 ∞ H P n + m , β ( α ) ( x , y , z ; k , a , b ) t n n ! w m m ! .$

Taking into account (1.1) and (4.3), then using (4.2), we get

$∑ n , m = 0 ∞ H P n + m , β ( α ) ( X , Y , Z ; k , a , b ) t n n ! w m m ! = ∑ n , m = 0 ∞ H P n + m , β ( α ) ( x , y , z ; k , a , b ) t n n ! w m m ! ∑ r , l = 0 ∞ H r + l ( 3 ) ( X − x , Y − y , Z − z ) t r r ! w l l ! = ∑ n , m = 0 ∞ ∑ r , l = 0 n , m ( n r ) ( m l ) H r + l ( 3 ) ( X − x , Y − y , Z − z ) H P n + m − r − l ( α ) ( x , y , z ; k , a , b ) t n n ! w m m ! .$

Whence the result. □

Corollary 4.6 Let $k=a=b=1$ and $β=λ$. For all $c,d,m∈N$, $n∈ N 0$, $λ∈C$, we have the following summation formula between the Hermite-based generalized Apostol-Bernoulli polynomials and 3d-Hermite polynomials:

$B n + m ( α ) H ( X , Y , Z ; λ ) = ∑ k , l = 0 n , m ( n k ) ( m l ) H k + l ( 3 ) ( X − x , Y − y , Z − z ) H B n + m − k − l ( α ) ( x , y , z ; λ ) .$

Corollary 4.7 Let $k+1=−a=b=1$ and $β=λ$. For all $m∈N$, $n∈ N 0$, $λ∈C$, we have the following summation formula between the Hermite-based generalized Apostol-Euler polynomials and 3d-Hermite polynomials:

$E n + m ( α ) H ( X , Y , Z ; λ ) = ∑ k , l = 0 n , m ( n k ) ( m l ) H k + l ( 3 ) ( X − x , Y − y , Z − z ) H E n + m − k − l ( α ) ( x , y , z ; λ ) .$

Corollary 4.8 Let $k=−2a=b=1$ and $2β=λ$. For all $m∈N$, $n∈ N 0$, $λ∈C$, we have the following summation formula between the Hermite-based generalized Apostol-Genocchi polynomials and 3d-Hermite polynomials:

$G n + m ( α ) H ( X , Y , Z ; λ ) = ∑ k , l = 0 n , m ( n k ) ( m l ) H k + l ( 3 ) ( X − x , Y − y , Z − z ) H G n + m − k − l ( α ) ( x , y , z ; λ ) .$

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Acknowledgements

Dedicated to Professor Hari M Srivastava.

Author information

Correspondence to Mehmet Ali Özarslan.

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The author declares that they have no competing interests.

Author’s contributions

The author completed the paper himself. The author read and approved the final manuscript.

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