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

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.

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 fN 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 gπ on X. Namely, we can consider g as a function on X.

For k0 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)=(1λ) δ 0 , n (see [1–19]),
(1.4)

where δ n , k is the Kronecker symbol.

For rN, 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 n0.
(1.9)

From (1.8) and (1.9), we have

t n | x n =n! δ n , k (n,k0).
(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,k0. 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 n0.
(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 (n0).

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 rN, 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 n0.

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.

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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.

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Kim, D.S., Kim, T., Lee, SH. et al. Frobenius-Euler polynomials and umbral calculus in the p-adic case. Adv Differ Equ 2012, 222 (2012). https://doi.org/10.1186/1687-1847-2012-222

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Keywords

  • Frobenius-Euler polynomials
  • umbral calculus
  • p-adic integral