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Some new identities of Bernoulli, Euler and Hermite polynomials arising from umbral calculus

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Abstract

In this paper, we derive the identities of higher-order Bernoulli, Euler and Frobenius-Euler polynomials from the orthogonality of Hermite polynomials. Finally, we give some interesting and new identities of several special polynomials arising from umbral calculus.

MSC: 05A10, 05A19.

1 Introduction

The Hermite polynomials are defined by the generating function to be

e 2 x t t 2 = e H ( x ) t = n = 0 H n (x) t n n !
(1.1)

with the usual convention about replacing H n (x) by H n (x) (see [1]). In the special case, x=0, H n (0)= H n are called the nth Hermite numbers. From (1.1) we have

H n (x)= ( H + 2 x ) n = l = 0 n ( n l ) H n l x l 2 l .
(1.2)

Thus, by (1.2), we get

d k d x k H n (x)= 2 k ( n ) k H n k (x)= 2 k n ! ( n k ) ! H n k (x),
(1.3)

where ( x ) k =x(x1)(xk+1).

As is well known, the Bernoulli polynomials of order r are defined by the generating function to be

( t e t 1 ) r e x t = n = 0 B n ( r ) (x) t n n ! (rR).
(1.4)

In the special case, x=0, B n ( r ) (0)= B n ( r ) are called the n th Bernoulli numbers of order r (see [14]).

The Euler polynomials of order r are also defined by the generating function to be

( 2 e t + 1 ) r e x t = n = 0 E n ( r ) (x) t n n ! (rR).
(1.5)

In the special case, x=0, E n ( r ) (0)= E n ( r ) are called the nth Euler numbers of order r.

For λ(1)C, the Frobenius-Euler polynomials of order r are given by

( 1 λ e t λ ) r e x t = n = 0 H n ( r ) (x|λ) t n n ! (rR).
(1.6)

In the special case, x=0, H n ( r ) (0|λ)= H n ( r ) (λ) are called the nth Frobenius-Euler numbers of order r (see [116]).

The Stirling numbers of the first kind are defined by the generating function to be

( x ) n = k = 0 n S 1 (n,k) x k (see [11, 14]),
(1.7)

and the Stirling numbers of the second kind are given by

( e t 1 ) n =n! l = n S 2 (l,n) t l l ! (see [14]).
(1.8)

In [1] it is known that H 0 (x), H 1 (x),, H n (x) from an orthogonal basis for the space

P n = { p ( x ) Q [ x ] | deg p ( x ) n }
(1.9)

with respect to the inner product

p 1 ( x ) , p 2 ( x ) = e x 2 p 1 (x) p 2 (x)dx(see [1]).
(1.10)

For p(x) P n , let us assume that

p(x)= k = 0 n C k H k (x).
(1.11)

Then, from the orthogonality of Hermite polynomials and Rodrigues’ formula, we have

C k = 1 2 k k ! π e x 2 H k ( x ) p ( x ) d x = ( 1 ) k 2 k k ! π ( d k d x k e x 2 ) p ( x ) d x (see [1]) .
(1.12)

In particular, for p(x)= x m (m0), we easily get

( d n d x n e x 2 ) x m d x = { 0 if  n > m  or  n m  with  m n 0 ( mod 2 ) , ( 1 ) n m ! π 2 m n ( m n 2 ) ! if  n m  with  m n 0 ( mod 2 ) .
(1.13)

Let be the set of all formal power series in the variable t over with

F= { f ( t ) = k = 0 a k k ! t k | a k C } .
(1.14)

Let us assume that is the algebra of polynomials in the variable x over and that P is the vector space of all linear functionals on . L|p(x) denotes the action of the linear functional L on polynomials p(x), and we remind that the vector space structure on P is defined by

L + M | p ( x ) = L | p ( x ) + M | p ( x ) , c L | p ( x ) = c L | p ( x ) ,

where c is a complex constant (see [2, 11, 14]).

The formal power series

f(t)= k = 0 a k k ! t k F
(1.15)

defines a linear functional on by setting

f ( t ) | x n = a n for all n Z + =N{0}.
(1.16)

Thus, by (1.15) and (1.16), we get

t k | x n =n! δ n , k (n,k0),
(1.17)

where δ n , k is the Kronecker symbol (see [2, 11, 14]).

Let f L (t)= k = 0 L | x k k ! t k . By (1.16), we get

f L ( t ) | x n = L | x n ,n0.
(1.18)

Thus, by (1.18), we see that f L (t)=L. The map L f L (t) is a vector space isomorphism from P onto . Henceforth, will be thought of as both a formal power series and a linear functional. We call the umbral algebra. The umbral calculus is the study of umbral algebra (see [2, 11, 14]).

The order o(f(t)) of the nonzero power series f(t) is the smallest integer k for which the coefficient of t k does not vanish. A series f(t) having o(f(t))=1 is called a delta series, and a series f(t) having o(f(t))=0 is called an invertible series (see [2, 11, 14]). By (1.16) and (1.17), we see that e y t |p(x)=p(y). For f(t)F and p(x)P, we have

f(t)= k = 0 f ( t ) | x k k ! t k ,p(x)= k = 0 t k | p ( x ) k ! x k .
(1.19)

Let f(t),g(t)F and p(x)P. Then we easily see that

f ( t ) g ( t ) | p ( x ) = f ( t ) | g ( t ) p ( x ) = g ( t ) | f ( t ) p ( x ) .
(1.20)

From (1.19), we can derive the following equation:

p ( k ) (0)= t k | p ( x ) and 1 | p ( k ) ( x ) = p ( k ) (0).
(1.21)

Thus, by (1.21), we get

t k p(x)= p ( k ) (x)= d k p ( x ) d x k (see [2, 11, 14]).
(1.22)

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 g(t)f ( t ) k | S n (x)=n! δ n , k , where n,k0 (see [2, 11, 14]). The sequence S n (x) is called Sheffer sequence for (g(t),f(t)), which is denoted by S n (x)(g(t),f(t)). For f(t)F and p(x)P, we have

e y t 1 t | p ( x ) = 0 y p(u)du, e y t 1 | p ( x ) =p(y)p(0),
(1.23)

and

f ( t ) | x p ( x ) = f ( t ) | p ( x ) .
(1.24)

In this paper, we introduce the identities of several special polynomials which are derived from the orthogonality of Hermite polynomials. Finally, we give some new and interesting identities of the higher-order Bernoulli, Euler and Frobenius-Euler polynomials arising from umbral calculus.

2 Some identities of several special polynomials

From (1.5), we note that

( 2 e t + 1 ) r = ( 1 + e t 1 2 ) r = j = 0 ( r j ) ( e t 1 2 ) j .
(2.1)

By (2.1), we get

( 2 e t + 1 ) r e x t = j = 0 ( r j ) ( e t 1 2 ) j e x t = n = 0 ( j = 0 n ( r j ) ( e t 1 2 ) j x n ) t n n ! .
(2.2)

From (1.5) and (2.2), we have

E n ( r ) (x)= j = 0 n ( r j ) 2 j ( e t 1 ) j x n .
(2.3)

By (1.8) and (1.9), we get

( e t 1 ) j x n = k = 0 n j t k | ( e t 1 ) j x n k ! = k = 0 n j ( e t 1 ) j | t k x n k ! x k = j ! k = 0 n j ( n k ) ( e t 1 ) j | x n k j ! x k = j ! k = 0 n j ( n j ) S 2 ( n k , j ) x k = j ! k = j n ( n k ) S 2 ( k , j ) x n k .
(2.4)

Therefore, by (2.3) and (2.4), we obtain the following theorem.

Theorem 2.1 For n0, we have

E n ( r ) ( x ) = 0 j n j k n ( n k ) ( r j ) j ! 2 j S 2 ( k , j ) x n k = 0 k n ( n k ) [ 0 j k ( r j ) j ! 2 j S 2 ( k , j ) ] x n k .

By (1.5), we easily see that

E n ( r ) (x)= k = 0 n ( n k ) E k ( r ) x n k .
(2.5)

Therefore, by Theorem 2.1 and (2.5), we obtain the following corollary.

Corollary 2.2 For k0, we have

E k ( r ) = j = 0 k ( r j ) j ! 2 j S 2 (k,j).

Let us take p(x)= E n ( r ) (x) P n . Then, by (1.11), we get

E n ( r ) (x)= k = 0 n C k H k (x).
(2.6)

From (1.12), we can derive the computation of C k as follows:

C k = ( 1 ) k 2 k k ! π ( d k e x 2 d x k ) E n ( r ) (x)dx,
(2.7)

where

( d k e x 2 d x k ) E n ( r ) ( x ) d x = ( n ) ( ( n 1 ) ) ( ( n k + 1 ) ) e x 2 E n k ( r ) ( x ) d x = ( 1 ) k n ! ( n k ) ! e x 2 l = 0 n k ( n k l ) E n k l ( r ) x l d x = ( 1 ) k n ! ( n k ) ! l = 0 n k ( n k l ) E n k l ( r ) e x 2 x l d x = ( 1 ) k n ! π 0 l n k , l : even 1 ( n k l ) ! 2 l ( l 2 ) ! j = 0 n k l ( r j ) j ! 2 j S 2 ( n k l , j ) .
(2.8)

From (2.7) and (2.8), we can derive the following equation:

C k = n ! 0 l n k , l : even E n k l ( r ) k ! ( n k l ) ! 2 k + l ( l 2 ) ! = n ! 0 l n k , l : even j = 0 n k l ( r j ) j ! S 2 ( n k l , j ) k ! ( n k l ) ! 2 k + l + j ( l 2 ) ! .
(2.9)

Therefore, by Corollary 2.2, (2.6) and (2.9), we obtain the following theorem.

Theorem 2.3 For n0, we have

E n ( r ) ( x ) = n ! k = 0 n { 0 l n k , l : even E n k l ( r ) k ! ( n k l ) ! 2 k + l ( l 2 ) ! } H k ( x ) = n ! k = 0 n { 0 l n k , l : even j = 0 n k l ( r j ) j ! S 2 ( n k l , j ) k ! ( n k l ) ! 2 k + l + j ( l 2 ) ! } H k ( x ) .

By (1.4), we easily see that

( t e t 1 ) r = ( 1 + e t t 1 t ) r = j = 0 ( r j ) ( e t t 1 t ) j .
(2.10)

Thus, by (2.10), we get

( t e t 1 ) r e x t = n = 0 ( j = 0 n ( r j ) ( e t t 1 t ) j x n ) t n n ! .
(2.11)

From (1.4) and (2.11), we have

B n ( r ) (x)= j = 0 n ( r j ) ( e t t 1 t ) j x n .
(2.12)

By (1.19), we easily get

( e t t 1 t ) j x n = k = 0 n j t k | ( e t t 1 t ) j x n k ! x k = k = 0 n j ( e t t 1 t ) j | t k x n k ! x k = k = 0 n j ( n k ) l = 0 j ( j l ) ( 1 ) j l ( e t 1 t ) l | x n k x k = k = 0 n j ( n k ) l = 0 j ( j l ) ( 1 ) j l t 0 | ( e t 1 t ) l x n k x k .
(2.13)

From (1.8), (1.21) and (2.13), we have

( e t t 1 t ) j x n = k = 0 n j l = 0 j ( n k ) ( j l ) ( 1 ) j l ( n k ) ! l ! ( n k + l ) ! S 2 (nk+l,l) x k .
(2.14)

Thus, by (2.12) and (2.14), we get

B n ( r ) ( x ) = j = 0 n k = 0 n j l = 0 j ( r j ) ( n k ) ( j l ) ( 1 ) j l S 2 ( n k + l , l ) ( n k + l l ) x k = k = 0 n ( n k ) [ j = 0 k l = 0 j ( r j ) ( j l ) S 2 ( k + l , l ) ( k + l l ) ( 1 ) j l ] x n k .
(2.15)

Therefore, by (2.12) and (2.15), we obtain the following theorem.

Theorem 2.4 For n0, we have

B n ( r ) (x)= k = 0 n ( n k ) [ j = 0 k l = 0 j ( r j ) ( j l ) S 2 ( k + l , l ) ( k + l l ) ( 1 ) j l ] x n k .

By (1.4), we easily get

B n ( r ) (x)= k = 0 n ( n k ) B k ( r ) x n k .
(2.16)

Therefore, by Theorem 2.4 and (2.16), we obtain the following corollary.

Corollary 2.5 For k0, we have

B k ( r ) = j = 0 k l = 0 j ( 1 ) j l ( r j ) ( j l ) S 2 ( k + l , l ) ( k + l l ) .

Let us consider p(x)= B n ( r ) (x) P n . Then, by (1.11), B n ( r ) (x) can be written as

B n ( r ) (x)= k = 0 n C k H k (x).
(2.17)

Now, we compute C k ’s for B k ( r ) (x) as follows:

C k = ( 1 ) k 2 k k ! π ( d k e x 2 d x k ) B n ( r ) (x)dx,
(2.18)

where

( d k e x 2 d x k ) B n ( r ) ( x ) d x = ( n ) ( ( n 1 ) ) ( ( n k + 1 ) ) e x 2 B n k ( r ) ( x ) d x = ( 1 ) k n ! ( n k ) ! l = 0 n k ( n k l ) B n k l ( r ) e x 2 x l d x = ( 1 ) k n ! π 0 l n k , l : even B n k l ( r ) ( n k l ) ! 2 l ( l 2 ) ! .
(2.19)

By Corollary 2.5 and (2.19), we get

( d k e x 2 d x k ) B n ( r ) ( x ) d x = ( 1 ) k n ! π 0 l n k , l : even s j = 0 n k l m = 0 j ( 1 ) j m ( r j ) ( j m ) S 2 ( n k l + m , m ) ( n k l ) ! 2 l ( l 2 ) ! ( n k l + m m ) .
(2.20)

From (2.18) and (2.20), we have

C k = n ! 0 l n k , l : even B n k l ( r ) ( n k l ) ! k ! 2 k + l ( l 2 ) ! = n ! 0 l n k , l : even j = 0 n k l m = 0 j ( 1 ) j m ( r j ) ( j m ) S 2 ( n k l + m , m ) ( n k l ) ! k ! 2 k + l ( l 2 ) ! ( n k l + m m ) .
(2.21)

Therefore, by (2.17) and (2.21), we obtain the following theorem.

Theorem 2.6 For n0, we have

B n ( r ) ( x ) = n ! k = 0 n { 0 l n k , l : even B n k l ( r ) ( n k l ) ! k ! 2 k + l ( l 2 ) ! } H k ( x ) = n ! k = 0 n { 0 l n k , l : even j = 0 n k l m = 0 j ( 1 ) j m ( r j ) ( j m ) S 2 ( n k l + m , m ) ( n k l ) ! k ! 2 k + l ( l 2 ) ! ( n k l + m m ) } H k ( x ) .

It is easy to show that

( 1 λ e t λ ) r = ( 1 + e t 1 1 λ ) r = j = 0 ( r j ) ( 1 1 λ ) j ( e t 1 ) j .
(2.22)

From (1.6) and (2.22), we have

H n ( r ) (x|λ)= j = 0 n ( r j ) ( 1 λ ) j ( e t 1 ) j x n ,
(2.23)

where

( e t 1 ) j x n = j ! k = j S 2 ( k , j ) t k k ! x n = j ! k = j n ( n k ) S 2 ( k , j ) x n k .
(2.24)

Thus, by (2.24), we get

( e t 1 ) j x n =j! k = j n ( n k ) S 2 (k,j) x n k .
(2.25)

From (2.23) and (2.25), we can derive the following equation:

H n ( r ) ( x | λ ) = j = 0 n k = j n ( n k ) ( r j ) j ! ( 1 λ ) j S 2 ( k , j ) x n k = k = 0 n j = 0 k ( n k ) ( r j ) j ! ( 1 λ ) j S 2 ( k , j ) x n k = k = 0 n ( n k ) [ j = 0 k ( r j ) j ! ( 1 λ ) j S 2 ( k , j ) ] x n k .
(2.26)

By (1.6), we easily see that

H n ( r ) (x|λ)= k = 0 n ( n k ) H k ( r ) (λ) x n k .
(2.27)

Therefore, by (2.26) and (2.27), we obtain the following theorem.

Theorem 2.7 For k0, we have

H k ( r ) (λ)= j = 0 k ( r j ) j ! ( 1 λ ) j S 2 (k,j).

Let us take p(x)= H n ( r ) (x|λ) P n . Then, by (1.11), H n ( r ) (x|λ) is given by

H n ( r ) (x|λ)= k = 0 n C k H k (x).
(2.28)

By (1.12), we get

C k = ( 1 ) k 2 k k ! π ( d k e x 2 d x k ) H n ( r ) (x|λ)dx,
(2.29)

where

( d k e x 2 d x k ) H n ( r ) ( x | λ ) d x = ( 1 ) k n ! ( n k ) ! l = 0 n k ( n k l ) H n k l ( r ) ( λ ) e x 2 x l d x = ( 1 ) k n ! π 0 l n k , l : even H n k l ( r ) ( λ ) ( n k l ) ! 2 l ( l 2 ) ! = ( 1 ) k n ! π 0 l n k , l : even j = 0 n k l ( r j ) j ! S 2 ( n k l , j ) ( n k l ) ! 2 l ( 1 λ ) j ( l 2 ) ! .
(2.30)

By (2.29) and (2.30), we get

C k = n ! 0 l n k , l : even H n k l ( r ) ( λ ) ( n k l ) ! k ! 2 l + k ( l 2 ) ! = n ! 0 l n k , l : even j = 0 n k l ( r j ) j ! S 2 ( n k l , j ) ( n k l ) ! k ! 2 k + l ( 1 λ ) j ( l 2 ) ! .
(2.31)

Therefore, by (2.28) and (2.31), we obtain the following theorem.

Corollary 2.8 For n0, we have

H n ( r ) ( x | λ ) = n ! k = 0 n { 0 l n k , l : even H n k l ( r ) ( λ ) ( n k l ) ! k ! 2 l + k ( l 2 ) ! } H k ( x ) = n ! k = 0 n { 0 l n k , l : even j = 0 n k l ( r j ) j ! S 2 ( n k l , j ) ( n k l ) ! k ! 2 k + l ( 1 λ ) j ( l 2 ) ! } H k ( x ) .

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Acknowledgements

The authors would like to express their sincere gratitude to referees for their valuable comments. This paper is supported in part by the Research Grant of Kwangwoon University in 2013.

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Correspondence to Taekyun Kim.

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Keywords

  • Bernoulli polynomial
  • Euler polynomial
  • Abel polynomial