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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. [1] 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 [2]. Note that the family of polynomials P n , β ( 1 ) (x,y,z;k,a,b) was investigated in [3].

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 αC, 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 , kN and ( β a ) b 1 (or 1); x,y,zR, βC, a,bC/{0}; 1 α :=1;

  2. (ii)

    |t|<π when ( β a ) b =1; |t|<|blog( β a )| when ( β a ) b 1; x,y,zR, k=0, α,βC, a,bC/{0}; 1 α :=1;

  3. (iii)

    |t|<π when α N 0 and ( β a ) b =1; x,y,zR, kN, βC, a,bC/{0}; 1 α :=1,

where w=|w| e i θ , πθ<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,zR. The Hermite-based generalized Apostol-Bernoulli polynomials are defined by

( t λ e t 1 ) α exp ( x t + y t 2 + z t 3 ) = n = 0 H B n ( α ) ( x , y , z ; λ ) t n n ! ( | t | < 2 π  when  α C  and  λ = 1 ; | t | < | log ( λ ) | when  α N 0  and  λ 1 ; x , y , z R ; 1 α : = 1 ) .

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 [47]). 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,zR. The Hermite-based generalized Apostol-Euler polynomials are defined by

( 2 λ e t + 1 ) α exp ( x t + y t 2 + z t 3 ) = n = 0 H E n ( α ) ( x , y , z ; λ ) t n n ! ( | t | < π  when  λ = 1 ; | t | < | log ( λ ) |  when  λ 1 ; x , y , z R , α C ; 1 α : = 1 ) .

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 [8]). 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,zR. The Hermite-based generalized Apostol-Genocchi polynomials are defined by

( 2 t λ e t + 1 ) α exp ( x t + y t 2 + z t 3 ) = n = 0 H G n ( α ) ( x , y , z ; λ ) t n n ! ( | t | < π  when  α N 0  and  λ = 1 ; | t | < | log ( λ ) | when  α N 0  and  λ 1 ; x , y , z R ; 1 α : = 1 ) .

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 [1125] and [26]. 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,bR{0}; βC 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 nN, 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 nN, 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 nN, 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 nN, 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 [27], 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 [27], 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 [2832] and [27], when y=z=0.

Theorem 3.1 Let c,d,mN, 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 [[33], 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,mN, n N 0 , λC, 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 mN, n N 0 , λC, 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 mN, n N 0 , λC, 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 [[34], p.101, Lemma 3])

n = 0 l = 0 A(n,l)= n = 0 l = 0 n A(nl,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 mN, n N 0 , λC, 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 mN, n N 0 , λC, 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 mN, n N 0 , λC, 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,mN, n N 0 , λC, 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 mN, n N 0 , λC, 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 mN, n N 0 , λC, 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 ) .

References

  1. 1.

    Khan S, Yasmin G, Khan R, Hassan NAM: Hermite-based Appell polynomials: properties and applications. J. Math. Anal. Appl. 2009, 351: 756–764. 10.1016/j.jmaa.2008.11.002

  2. 2.

    Özarslan MA: Unified Apostol-Bernoulli, Euler and Genocchi polynomials. Comput. Math. Appl. 2011, 62(6):2452–2462. 10.1016/j.camwa.2011.07.031

  3. 3.

    Ozden H, Simsek Y, Srivastava HM: A unified presentation of the generating functions of the generalized Bernoulli, Euler and Genocchi polynomials. Comput. Math. Appl. 2010, 60(10):2779–2787. 10.1016/j.camwa.2010.09.031

  4. 4.

    Luo Q-M: On the Apostol-Bernoulli polynomials. Cent. Eur. J. Math. 2004, 2(4):509–515. 10.2478/BF02475959

  5. 5.

    Luo Q-M, Srivastava HM: Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials. Comput. Math. Appl. 2006, 51(3–4):631–642. 10.1016/j.camwa.2005.04.018

  6. 6.

    Luo Q-M, Srivastava HM: Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials. J. Math. Anal. Appl. 2005, 308(1):290–302. 10.1016/j.jmaa.2005.01.020

  7. 7.

    Srivastava HM: Some formulas for the Bernoulli and Euler polynomials at rational arguments. Math. Proc. Camb. Philos. Soc. 2000, 129(1):77–84. 10.1017/S0305004100004412

  8. 8.

    Luo Q-M: Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions. Taiwan. J. Math. 2006, 10: 917–925.

  9. 9.

    Luo Q-M: Extension for the Genocchi polynomials and its Fourier expansions and integral representations. Osaka J. Math. 2011, 48(2):291–309.

  10. 10.

    Luo Q-M: Fourier expansions and integral representations for the Genocchi polynomials. J. Integer Seq. 2009., 12: Article ID 09.1.4

  11. 11.

    Dere R, Simsek Y: Genocchi polynomials associated with the umbral algebra. Appl. Math. Comput. 2011, 218: 756–761. 10.1016/j.amc.2011.01.078

  12. 12.

    Dere R, Simsek Y: Applications of umbral algebra to some special polynomials. Adv. Stud. Contemp. Math. 2012, 22: 433–438.

  13. 13.

    Karande BK, Thakare NK: On the unification of Bernoulli and Euler polynomials. Indian J. Pure Appl. Math. 1975, 6: 98–107.

  14. 14.

    Kurt B, Simsek Y: Frobenius-Euler type polynomials related to Hermite-Bernoulli polynomials. Analysis and applied math. AIP Conf. Proc. 2011, 1389: 385–388.

  15. 15.

    Luo Q-M: q -Extensions for the Apostol-Genocchi polynomials. Gen. Math. 2009, 17: 113–125.

  16. 16.

    Luo Q-M: The multiplication formulas for the Apostol-Bernoulli and Apostol-Euler polynomials of higher order. Integral Transforms Spec. Funct. 2009, 20(5–6):377–391.

  17. 17.

    Luo Q-M, Srivastava HM: Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind. Appl. Math. Comput. 2011, 217: 5702–5728. 10.1016/j.amc.2010.12.048

  18. 18.

    Ozden H, Simsek Y: A new extension of q -Euler numbers and polynomials related to their interpolation functions. Appl. Math. Lett. 2008, 21: 934–939. 10.1016/j.aml.2007.10.005

  19. 19.

    Simsek Y:Complete sum of products of (h,q)-extension of Euler polynomials and numbers. J. Differ. Equ. Appl. 2010, 16(11):1331–1348. 10.1080/10236190902813967

  20. 20.

    Simsek Y: Twisted (h,q) -Bernoulli numbers and polynomials related to twisted (h,q) -zeta function and L -function. J. Math. Anal. Appl. 2006, 324: 790–804. 10.1016/j.jmaa.2005.12.057

  21. 21.

    Simsek Y: Twisted p -adic (h,q) - L -functions. Comput. Math. Appl. 2010, 59: 2097–2110. 10.1016/j.camwa.2009.12.015

  22. 22.

    Srivastava HM: Some generalizations and basic (or q -) extensions of the Bernoulli, Euler and Genocchi polynomials. Appl. Math. Inf. Sci. 2011, 5: 390–444.

  23. 23.

    Srivastava HM, Choi J: Series Associated with the Zeta and Related Functions. Kluwer Academic, Dordrecht; 2001.

  24. 24.

    Srivastava HM, Choi J: Zeta and q-Zeta Functions and Associated Series and Integrals. Elsevier, Amsterdam; 2012.

  25. 25.

    Srivastava HM, Garg M, Choudhary S: A new generalization of the Bernoulli and related polynomials. Russ. J. Math. Phys. 2010, 17: 251–261. 10.1134/S1061920810020093

  26. 26.

    Srivastava HM, Özarslan MA, Kaanoğlu C: Some generalized Lagrange-based Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials. Russ. J. Math. Phys. 2013, 20: 110–120.

  27. 27.

    Zhang Z, Yang H: Several identities for the generalized Apostol-Bernoulli polynomials. Comput. Math. Appl. 2008, 56(12):2993–2999. 10.1016/j.camwa.2008.07.038

  28. 28.

    Deeba E, Rodriguez D: Stirling’s series and Bernoulli numbers. Am. Math. Mon. 1991, 98: 423–426. 10.2307/2323860

  29. 29.

    Kurt V: A further symmetric relation on the analogue of the Apostol-Bernoulli and the analogue of the Apostol-Genocchi polynomials. Appl. Math. Sci. 2009, 3(53–56):2757–2764.

  30. 30.

    Raabe JL: Zurückführung einiger Summen und bestimmten Integrale auf die Jakob Bernoullische Function. J. Reine Angew. Math. 1851, 42: 348–376.

  31. 31.

    Tuenter HJH: A symmetry of power sum polynomials and Bernoulli numbers. Am. Math. Mon. 2001, 108: 258–261. 10.2307/2695389

  32. 32.

    Yang SL: An identity of symmetry for the Bernoulli polynomials. Discrete Math. 2008, 308(4):550–554. 10.1016/j.disc.2007.03.030

  33. 33.

    Srivastava HM, Özarslan MA, Kaanoğlu C: Some families of generating functions for a certain class of three-variable polynomials. Integral Transforms Spec. Funct. 2010, 21(12):885–896. 10.1080/10652469.2010.481439

  34. 34.

    Srivastava HM, Manocha HL: A Treatise on Generating Functions. Halsted, New York; 1984.

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Acknowledgements

Dedicated to Professor Hari M Srivastava.

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Correspondence to Mehmet Ali Özarslan.

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

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Keywords

  • Hermite-based Apostol-Bernoulli polynomials
  • Hermite-based Apostol-Euler polynomials
  • Hermite-based Apostol-Genocchi polynomials
  • generalized sum of integer powers
  • generalized sum of alternative integer powers