Open Access

Frobenius-Euler polynomials and umbral calculus in the p-adic case

Advances in Difference Equations20122012:222

https://doi.org/10.1186/1687-1847-2012-222

Received: 21 November 2012

Accepted: 6 December 2012

Published: 20 December 2012

Abstract

In this paper, we study some p-adic Frobenius-Euler measure related to umbral calculus in the p-adic case. Finally, we derive some identities of Frobenius-Euler polynomials from our study.

MSC:05A10, 05A19.

Keywords

Frobenius-Euler polynomialsumbral calculusp-adic integral

1 Introduction

Let p be a fixed prime number. Throughout this paper Z p , Q p and C p will denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of Q p , respectively.

For f N with ( f , p ) = 1 , let
Note that the natural map induces
π : X Z p .

If g is a function on Z p , we denote by the same g the function g π on X. Namely, we can consider g as a function on X.

For k 0 and λ C p with | 1 λ | p > 1 , the Frobenius-Euler measure on X is defined by
μ λ ( x + f p N Z p ) = λ f p N x 1 λ f p N ( see [5, 8]) ,
(1.1)

where the p-adic absolute value on C p is normalized by | p | p = 1 p .

As is well known, the Frobenius-Euler polynomials are defined by the generating function to be
( 1 λ e t λ ) e x t = e H ( x | λ ) t = n = 0 H n ( x | λ ) t n n ! ( see [5, 7, 9]) ,
(1.2)
with the usual convention about replacing H n ( x | λ ) by H n ( x | λ ) . In the special case, x = 0 , H n ( 0 | λ ) = H n ( λ ) are called the n th Frobenius-Euler numbers
H n ( x | λ ) = ( H ( λ ) + x ) n = l = 0 ( n l ) H l ( λ ) x n l (see [6, 9, 10]) .
(1.3)
Thus, by (1.2) and (1.3), we easily get
( H ( λ ) + 1 ) n λ H n ( λ ) = ( 1 λ ) δ 0 , n ( see [1–19]) ,
(1.4)

where δ n , k is the Kronecker symbol.

For r N , the Frobenius-Euler polynomials of order r are defined by the generating function
( 1 λ e t λ ) r e x t = ( 1 λ e t λ ) × × ( 1 λ e t λ ) r -times e x t = n = 0 H n ( r ) ( x | λ ) t n n ! ( see [5, 9]) .
(1.5)
In the special case, x = 0 , H n ( r ) ( 0 | λ ) = H n ( r ) ( λ ) are called the n th Frobenius-Euler numbers of order r. The n th Frobenius-Euler polynomials can be represented by (1.1) as follows:
λ H n ( x | λ ) 1 λ = X ( x + y ) n d μ λ ( y ) = Z p ( x + y ) n d μ λ ( y ) = lim N 1 1 λ p N y = 0 p N 1 ( x + y ) n λ p N y ( see [6, 7]) .
(1.6)
Let be the set of all formal power series in the variable t over C p with
F = { f ( t ) = k = 0 a k k ! t k | a k C p } .
(1.7)

Let P = C p [ x ] and P denote the vector space of all linear functionals on .

The formal power series
f ( t ) = k = 0 a k k ! t k F ( see [11, 15])
(1.8)
defines a linear functional on by setting
f ( t ) | x n = a n for all  n 0 .
(1.9)
From (1.8) and (1.9), we have
t n | x n = n ! δ n , k ( n , k 0 ) .
(1.10)

Here, denotes both the algebra of formal power series in t and the vector space of all linear functionals on , and so an element f ( t ) of will be thought of as both a formal power series and a linear functional (see [11, 15]). We will call the umbral algebra. The umbral calculus is the study of umbral algebra (see [11, 15]).

The order o ( f ( t ) ) of power series f ( t ) (≠0) is the smallest integer k for which a k does not vanish (see [11, 15]). A series f ( t ) for which o ( f ( t ) ) = 1 is called a delta series. If a series f ( t ) has o ( f ( t ) ) = 0 , then f ( t ) is called an invertible series (see [11, 15]). Let f ( t ) , g ( t ) F . Then we easily see that f ( t ) g ( t ) | p ( x ) = f ( t ) | g ( t ) p ( x ) = g ( t ) | f ( t ) p ( x ) . From (1.10), we note that
e y t | x n = y n , e y t | p ( x ) = p ( y ) ,
(1.11)
f ( t ) = k = 0 f ( t ) | x k k ! t k , f ( t ) F ,
(1.12)
and
p ( x ) = k = 0 t k | p ( x ) k ! x k , p ( x ) P  (see [15]) .
(1.13)
For f 1 ( t ) , f 2 ( t ) , , f m ( t ) F , we have
f 1 ( t ) f 2 ( t ) f m ( t ) | x n = i 1 + + i m = n ( n i 1 , , i m ) f 1 ( t ) | x i 1 f m ( t ) | x i m .
(1.14)
By (1.13), we get
p ( k ) ( x ) = d k p ( x ) d x k = l = k n t l | p ( x ) ( l k ) k ! l ! x l k
(1.15)
and
p ( k ) ( 0 ) = t k | p ( x ) = 1 | p ( k ) ( x ) .
Thus, by (1.15), we get
t k p ( x ) = p ( k ) ( x ) = d k p ( x ) d x k (see [11, 15]) .
(1.16)
By (1.16), we easily see that
e y t p ( x ) = p ( x + y ) (see [15]) .
(1.17)

Let S n ( x ) denote a polynomial of degree n. Suppose that f ( t ) , g ( t ) F with o ( f ( t ) ) = 1 and o ( g ( t ) ) = 0 . Then there exists a unique sequence S n ( x ) of polynomials satisfying 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 ) ) , which is denoted by S n ( x ) ( g ( t ) , f ( t ) ) . If S n ( x ) ( g ( t ) , t ) , then S n ( x ) is called the Appell sequence for g ( t ) (see [15]).

For p ( x ) P , we have
f ( t ) | x p ( x ) = t f ( t ) | p ( x ) = f ( t ) | p ( x ) ,
(1.18)
e y t 1 | p ( x ) = p ( y ) p ( 0 ) .
(1.19)
If S n ( x ) ( g ( t ) , f ( t ) ) , then we have
h ( t ) = k = 0 h ( t ) | S k ( x ) k ! g ( t ) f ( t ) k , h ( t ) F ,
(1.20)
p ( x ) = k = 0 g ( t ) f ( t ) k | p ( x ) k ! S k ( x ) , p ( x ) P ,
(1.21)
f ( t ) S n ( x ) = n S n 1 ( x ) ,
(1.22)
and
1 g ( f ¯ ( t ) ) e y f ¯ ( t ) = k = 0 S k ( y ) k ! t k for all  y C p ,
(1.23)

where f ¯ ( t ) is compositional inverse of f ( t ) (see [11, 15]). In [9], Kim and Kim have studied some identities of Frobenius-Euler polynomials arising from umbral calculus. In this paper, we study some p-adic Frobenius-Euler integral on related to umbral calculus in the p-adic case. Finally, we derive some new and interesting identities of Frobenius-Euler polynomials from our study.

2 Frobenius-Euler polynomials associated with umbral calculus

Let
g ( t ; λ ) = e t λ 1 λ F .
(2.1)
Then we see that g ( t ; λ ) is an invertible series. From (1.2), we have
k = 0 H k ( x | λ ) t k k ! = 1 g ( t ; λ ) e x t .
(2.2)
Hence, by (2.2), we get
( 1 λ e t λ ) x n = 1 g ( t ; λ ) x n = H n ( x | λ ) .
(2.3)
By (2.2) and (2.3), we get
H n ( x | λ ) ( g ( t ; λ ) , t ) .
From (1.6), we have
Z p e ( x + y ) t d μ λ ( y ) = λ e t λ e x t ,
(2.4)
and
Z p e ( x + y + 1 ) t d μ λ ( y ) λ Z p e ( x + y ) t d μ λ ( y ) = λ e x t .
(2.5)
By (2.5), we get
Z p ( x + y + 1 ) n d μ λ ( y ) λ Z p ( x + y ) n d μ λ ( y ) = λ x n .
(2.6)
From (1.6) and (2.6), we have
λ 1 λ H n ( x + 1 | λ ) λ 2 1 λ H n ( x | λ ) = λ x n .
(2.7)
From (2.2), we can easily derive
H n + 1 ( x | λ ) = ( x g ( t ; λ ) g ( t ; λ ) ) H n ( x | λ ) .
(2.8)
By (2.8), we get
g ( t ; λ ) H n + 1 ( x | λ ) = g ( t ; λ ) x H n ( x | λ ) g ( t ; λ ) H n ( x | λ ) .
(2.9)
Thus, from (2.9), we have
( e t λ ) H n + 1 ( x | λ ) = ( e t λ ) x H n ( x | λ ) e t H n ( x | λ ) .
(2.10)
By (2.10), we get
H n + 1 ( x + 1 | λ ) λ H n + 1 ( x | λ ) = x ( H n ( x + 1 | λ ) λ H n ( x | λ ) ) .
(2.11)
From (2.11), we note that
H n ( x + 1 | λ ) λ H n ( x | λ ) = x ( H n 1 ( x + 1 | λ ) λ H n 1 ( x | λ ) ) = x 2 ( H n 2 ( x + 1 | λ ) λ H n 2 ( x | λ ) ) = = x n ( H 0 ( x + 1 | λ ) λ H 0 ( x | λ ) ) = x n ( 1 λ ) .
(2.12)
Let us consider the linear functional f ( t ) such that
f ( t ) | p ( x ) = Z p p ( u ) d μ λ ( u )
(2.13)
for all polynomials p ( x ) can be determined from (1.12) to be
f ( t ) = k = 0 f ( t ) | x k k ! t k = k = 0 Z p u k d μ λ ( u ) t k k ! = Z p e u t d μ λ ( u ) .
(2.14)
By (2.4) and (2.14), we get
f ( t ) = Z p e u t d μ λ ( u ) = λ e t λ .
(2.15)

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

Theorem 2.1 For p ( x ) P , we have
λ e t λ | p ( x ) = Z p p ( u ) d μ λ ( u ) .
In particular,
λ 1 λ H n ( λ ) = Z p e y t d μ λ ( y ) | x n .
From (1.6), we have
n = 0 Z p ( x + y ) n d μ λ ( y ) t n n ! = Z p e ( x + y ) t d μ λ ( y ) = n = 0 Z p e y t d μ λ ( y ) x n t n n ! .
(2.16)
By (1.6), (2.4) and (2.16), we get
λ 1 λ H n ( x | λ ) = Z p e y t d μ λ ( y ) x n = λ e t λ x n , for  n 0 .
(2.17)

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

Theorem 2.2 For p ( x ) P , we have
Z p p ( x + y ) d μ λ ( y ) = Z p e y t d μ λ ( y ) p ( x ) = λ e t λ p ( x ) .
In particular,
λ 1 λ H n ( x | λ ) = Z p e y t d μ λ ( y ) x n = λ e t λ x n ( n 0 ) .
By (1.6) and (2.16), we get
λ 1 λ H n ( x | λ ) ( e t λ λ , t ) .
(2.18)
From Appell identity and (2.18), we can derive the following identities:
H n ( x + y | λ ) = k = 0 n ( n k ) H k ( x | λ ) y n k .
(2.19)
Let
g r ( t ; λ ) = ( e t λ λ ) r = ( e t λ λ ) × × ( e t λ λ ) r -times F .
(2.20)
Then g r ( t ; λ ) is an invertible functional in . By (1.5) and (2.20), we get
1 g r ( t ; λ ) e x t = λ r ( 1 λ ) r k = 0 H n ( r ) ( x | λ ) t n n ! .
(2.21)
Thus, from (2.21), we have
1 g r ( t ; λ ) x n = ( λ 1 λ ) r H n ( r ) ( x | λ ) ,
(2.22)
and
( λ 1 λ ) r t H n ( r ) ( x | λ ) = n g r ( t ; λ ) x n 1 = n ( λ 1 λ ) r H n 1 ( r ) ( x | λ ) .
(2.23)
By (2.22) and (2.23), we see that
( λ 1 λ ) r H n ( r ) ( x | λ ) ( g r ( t ; λ ) , t ) .
(2.24)
From (2.4), we can derive the following identity:
Z p Z p r -times e ( x 1 + x 2 + + x r + x ) t d μ λ ( x 1 ) d μ λ ( x r ) = ( λ e t λ ) r e x t = ( λ 1 λ ) r n = 0 H n ( r ) ( x | λ ) t n n ! .
(2.25)
By (1.10) and (2.25), we get
( λ 1 λ ) r H n ( r ) ( x | λ ) = Z p Z p r -times e ( x 1 + x 2 + + x r + x ) t d μ λ ( x 1 ) d μ λ ( x r ) | x n .
(2.26)
From (1.14), we have
Z p Z p r -times e ( x 1 + x 2 + + x r ) t d μ λ ( x 1 ) d μ λ ( x r ) | x n = n = i 1 + + i r ( n i 1 , , i r ) Z p e x 1 t d μ λ ( x 1 ) | x i 1 × × Z p e x r t d μ λ ( x r ) | x i r = n = i 1 + + i r ( n i 1 , , i r ) ( λ 1 λ ) r H i 1 ( x | λ ) H i r ( x | λ ) .
(2.27)
By (2.26) and (2.27), we get
H n ( r ) ( x | λ ) = n = i 1 + + i r ( n i 1 , , i r ) H i 1 ( x | λ ) H i r ( x | λ ) ,
where ( n i 1 , , i r ) = n ! i 1 ! i r ! . From (2.25), we note that
g r ( t ; λ ) = 1 Z p Z p r -times e ( x 1 + x 2 + + x r ) t d μ λ ( x 1 ) d μ λ ( x r ) = ( e t λ λ ) r .
(2.28)
Thus, by (2.28), we get
1 g r ( t ; λ ) e x t = Z p Z p e ( x 1 + x 2 + + x r ) t d μ λ ( x 1 ) d μ λ ( x r ) e x t = Z p Z p e ( x 1 + x 2 + + x r + x ) t d μ λ ( x 1 ) d μ λ ( x r ) = ( λ 1 λ ) r n = 0 H n ( r ) ( x | λ ) t n n ! .
(2.29)
By (2.29), we see that
( λ 1 λ ) r H n ( r ) ( x | λ ) = Z p Z p r -times ( x + x 1 + + x r ) n d μ λ ( x 1 ) d μ λ ( x r ) = Z p Z p r -times e ( x 1 + x 2 + + x r ) t d μ λ ( x 1 ) d μ λ ( x r ) x n = 1 g r ( t ; λ ) x n .
(2.30)

Therefore, by (2.30), we obtain the following theorem.

Theorem 2.3 For p ( x ) P and r N , we have
Z p Z p r -times p ( x 1 + x 2 + + x r + x ) d μ λ ( x 1 ) d μ λ ( x r ) = ( λ e t λ ) r p ( x ) .
In particular,
( λ 1 λ ) r H n ( r ) ( x | λ ) = Z p Z p e ( x 1 + x 2 + + x r ) t d μ λ ( x 1 ) d μ λ ( x r ) x n .
Moreover,
( λ 1 λ ) r H n ( r ) ( x | λ ) ( 1 Z p Z p e ( x 1 + x 2 + + x r ) t d μ λ ( x 1 ) d μ λ ( x r ) , t ) .
Let us consider the function f ( t ) in such that
f ( t ) | p ( x ) = Z p Z p p ( x 1 + x 2 + + x r ) d μ λ ( x 1 ) d μ λ ( x r )
(2.31)
for all polynomials p ( x ) can be determined from (1.12) to be
f ( t ) = k = 0 f ( t ) | x k k ! t k = k = 0 Z p Z p r -times ( x 1 + + x r ) k d μ λ ( x 1 ) d μ λ ( x r ) t k k ! = Z p Z p r -times e ( x 1 + + x r ) t d μ λ ( x 1 ) d μ λ ( x r ) .
(2.32)

Therefore, by (2.31) and (2.32), we obtain the following theorem.

Theorem 2.4 For p ( x ) P , we have
Z p Z p r -times e ( x 1 + x 2 + + x r ) t d μ λ ( x 1 ) d μ λ ( x r ) | p ( x ) = Z p Z p r -times p ( x 1 + x 2 + + x r ) d μ λ ( x 1 ) d μ λ ( x r ) .
In particular,
( λ e t λ ) r | p ( x ) = Z p Z p r -times p ( x 1 + x 2 + + x r ) d μ λ ( x 1 ) d μ λ ( x r ) .
Indeed, the n th Frobenius-Euler number of order r is given by
( λ 1 λ ) r H n ( r ) ( x | λ ) = Z p Z p r -times e ( x 1 + x 2 + + x r ) t d μ λ ( x 1 ) d μ λ ( x r ) | x n ,

where n 0 .

Remark From (1.2) and (1.5), we note that
d d λ ( 1 λ e t λ ) = 1 e t ( e t λ ) 2 = 1 ( 1 λ ) ( ( 1 λ ) 2 ( e t λ ) 2 1 λ e t λ ) = 1 1 λ n = 0 ( H n ( 2 ) ( λ ) H n ( λ ) ) t n n ! ,
(2.33)
and
d 2 d λ 2 ( 1 λ e t λ ) = 2 ! 1 e t ( e t λ ) 3 = 2 ! ( 1 λ ) 2 ( ( 1 λ ) 3 ( e t λ ) 3 ( 1 λ ) 2 ( e t λ ) 2 ) = 2 ! ( 1 λ ) 2 n = 0 ( H n ( 3 ) ( λ ) H n ( 2 ) ( λ ) ) t n n ! .
(2.34)
Continuing this process, we obtain the following equation:
d k d λ k ( 1 λ e t λ ) = k ! ( 1 λ ) k ( ( 1 λ ) k + 1 ( e t λ ) k + 1 ( 1 λ ) k ( e t λ ) k ) = k ! ( 1 λ ) k n = 0 ( H n ( k + 1 ) ( λ ) H n ( k ) ( λ ) ) t n n ! .
(2.35)
By (1.2), (1.5) and (2.35), we get
d k d λ k H n ( λ ) = k ! ( 1 λ ) k ( H n ( k + 1 ) ( λ ) H n ( k ) ( λ ) ) ,

where k is a positive integer.

Declarations

Acknowledgements

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology 2012R1A1A2003786.

Authors’ Affiliations

(1)
Department of Mathematics, Sogang University
(2)
Department of Mathematics, Kwangwoon University
(3)
Division of General Education, Kwangwoon University
(4)
Department of Mathematics Education, Kyungpook National University

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