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Some identities of Frobenius-Euler polynomials arising from umbral calculus

Abstract

In this paper, we study some interesting identities of Frobenius-Euler polynomials arising from umbral calculus.

1 Introduction

Let C be the complex number field, and let F be the set of all formal power series in the variable t over C with

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

We use notation P=C[x] and P denotes the vector space of all linear functional on .

Also, L|p(x) denotes the action of the linear functional L on the polynomial p(x), and we remind that the vector space operations on P is defined by

L + M | p ( x ) = L | p ( x ) + M | p ( x ) , c L | p ( x ) = c L | p ( x ) ( see [1] ) ,

where c is any constant in C.

The formal power series

f(t)= k = 0 a k k ! t k F(see [1, 2]),
(1)

defines a linear functional on by setting

f ( t ) | x n = a n ,for all n0.
(2)

In particular,

t k | x n =n! δ n , k ,
(3)

where δ n , k is the Kronecker symbol. If f L (t)= k = 0 L | x k k ! t k , then we get f L (t)| x n =L| x n and so as linear functionals L= f L (t) (see [1, 2]).

In addition, the map L f L (t) is a vector space isomorphism from P onto F (see [1, 2]). Henceforth, F will denote 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 F will be thought of as both a formal power series and a linear functional. We shall call F the umbral algebra (see [1, 2]).

Let us give an example. For y in C the evaluation functional is defined to be the power series e y t . From (2), we have e y t | x n = y n and so e y t |p(x)=p(y) (see [1, 2]). Notice that for all f(t) in F,

f(t)= k = 0 f ( t ) | x t k ! t k
(4)

and for all polynomial p(x)

p(x)= k 0 t k | p ( x ) k ! x k (see [1, 2]).
(5)

For f 1 (t), f 2 (t),, f m (t)F, we have

f 1 ( t ) f 2 ( t ) f m ( t ) | x n = ( n i 1 , , i m ) f 1 ( t ) | x i 1 f n ( t ) | x i m ,

where the sum is over all nonnegative integers i 1 , i 2 ,, i m such that i 1 ++ i m =n (see [1, 2]). The order o(f(t)) of the power series f(t)0 is the smallest integer k for which a k does not vanish. We define o(f(t))= if f(t)=0. We see that o(f(t)g(t))=o(f(t))+o(g(t)) and o(f(t)+g(t))min{o(f(t)),o(g(t))}. The series f(t) has a multiplicative inverse, denoted by f ( t ) 1 or 1 f ( t ) , if and only if o(f(t))=0. Such series is called an invertible series. A series f(t) for which o(f(t))=1 is called a delta series (see [1, 2]). For f(t),g(t)F, we have f(t)g(t)|p(x)=f(t)|g(t)p(x).

A delta series f(t) has a compositional inverse f ¯ (t) such that f( f ¯ (t))= f ¯ (f(t))=t.

For f(t),g(t)F, we have f(t)g(t)|p(x)=f(t)|g(t)p(x).

From (5), we have

p ( k ) (x)= d k p ( x ) d x k = l = k t l | p ( x ) l ! l(l1)(lk+1) x l k .

Thus, we see that

p ( k ) (0)= t k | p ( x ) = 1 | p ( k ) ( x ) .
(6)

By (6), we get

t k p(x)= p ( k ) (x)= d k ( p ( x ) ) d x k (see [1, 2]).
(7)

By (7), we have

e y t p(x)=p(x+y)(see [1, 2]).
(8)

Let S n (x) be a polynomial with deg S n (x)=n.

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 such that 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)) or that S n (t) is Sheffer for (g(t),f(t)).

The Sheffer sequence for (1,f(t)) is called the associated sequence for f(t) or S n (x) is associated to f(t). The Sheffer sequence for (g(t),t) is called the Appell sequence for g(t) or S n (x) is Appell for g(t) (see [1, 2]). The umbral calculus is the study of umbral algebra and the modern classical umbral calculus can be described as a systemic study of the class of Sheffer sequences. Let p(x)P. Then we have

(9)
(10)

and

e y t 1 | p ( x ) =p(y)p(0)(see [1, 2]).
(11)

Let S n (x) be Sheffer for (g(t),f(t)). Then

(12)
(13)
(14)
(15)

For λ(1)C, we recall that 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 ! ,
(16)

with the usual convention about replacing H n (x|λ) by H n (x|λ) (see [3]). In the special case, x=0, H n (0|λ)= H n (λ) are called the n th Frobenius-Euler numbers. By (16), we get

H n (x|λ)= ( H ( λ ) + x ) n = l = 0 n ( n l ) H n l ( λ ) x l ,
(17)

and

( H ( λ ) + 1 ) n λ H n (λ)=(1λ) δ 0 , n (see [1, 4–13]).
(18)

From (17), we note that the leading coefficient of H n (x|λ) is H 0 (λ)=1. So, H n (x|λ) is a monic polynomial of degree n with coefficients in Q(λ).

In this paper, we derive some new identities of Frobenius-Euler polynomials arising from umbral calculus.

2 Applications of umbral calculus to Frobenius-Euler polynomials

Let S n (x) be an Appell sequence for g(t). From (14), we have

1 g ( t ) x n = S n (x)if and only if x n =g(t) S n (x)(n0).
(19)

For λ(1)C, let us take g λ (t)= e t λ 1 λ F.

Then we see that g λ (t) is an invertible series.

From (16), we have

k = 0 H k ( x | λ ) k ! t k = 1 g λ ( t ) e x t .
(20)

By (20), we get

1 g λ ( t ) x n = H n (x|λ) ( λ ( 1 ) C , n 0 ) ,
(21)

and by (17), we get

t H n (x|λ)= H n (x|λ)=n H n 1 (x|λ).
(22)

Therefore, by (21) and (22), we obtain the following proposition.

Proposition 1 For λ(1)C, n0, we see that H n (x|λ) is the Appell sequence for g λ (t)= e t λ 1 λ .

From (20), we have

k = 1 H k ( x | λ ) k ! k t k 1 = x g λ ( t ) e x t g λ ( t ) e x t g λ ( t ) 2 = k = 0 { x 1 g λ ( t ) x k g λ ( t ) g λ ( t ) 1 g λ ( t ) x k } t k k ! .
(23)

By (21) and (23), we get

H k + 1 (x|λ)=x H k (x|λ) g λ ( t ) g λ ( t ) H k (x|λ).
(24)

Therefore, by (24) we obtain the following theorem.

Theorem 2 Let g λ (t)= e t λ 1 λ F. Then we have

H k + 1 (x|λ)= ( x g λ ( t ) g λ ( t ) ) H k (x|λ)(k0).

From (16), we have

n = 0 ( H n ( x + 1 | λ ) λ H n ( x | λ ) ) t n n ! = 1 λ e t λ e ( x + 1 ) t λ 1 λ e t λ e x t =(1λ) e x t .
(25)

By (25), we get

H n (x+1|λ)λ H n (x|λ)=(1λ) x n .
(26)

From Theorem 2, we can derive the following equation (27):

g λ (t) H k + 1 (x|λ)= ( g λ ( t ) x g λ ( t ) ) H k (x|λ).
(27)

By (27), we get

( e t λ 1 λ ) H k + 1 (x|λ)= e t λ 1 λ x H k (x|λ) e t 1 λ H k (x|λ).
(28)

From (8) and (28), we have

H k + 1 ( x + 1 | λ ) λ H k + 1 ( x | λ ) = ( x + 1 ) H k ( x + 1 | λ ) λ x H k ( x | λ ) H k ( x + 1 | λ ) = x H k ( x + 1 | λ ) λ x H k ( x | λ ) .

Therefore, by (26), we obtain the following theorem.

Theorem 3 For k0, we have

H k + 1 (x+1|λ)=λ H k + 1 (x|λ)+(1λ) x k + 1 .

From (16), (17), and (18), we note that

x x + y H n ( u | λ ) d u = 1 n + 1 { H n + 1 ( x + y | λ ) H n + 1 ( x | λ ) } = 1 n + 1 k = 1 ( n + 1 k ) H n + 1 k ( x | λ ) y k = k = 1 n ( n 1 ) ( n k + 2 ) k ! H n + 1 k ( x | λ ) y k = k = 1 y k k ! t k 1 H n ( x | λ ) = 1 t ( k = 0 y k k ! t k 1 ) H n ( x | λ ) = e y t 1 t H n ( x | λ ) .
(29)

Therefore, by (29), we obtain the following theorem.

Theorem 4 For λ(1)C, n0, we have

x x + y H n (u|λ)du= e y t 1 t H n (x|λ).

By (15) and Proposition 1, we get

t { 1 n + 1 H n + 1 ( x | λ ) } = H n (x|λ).
(30)

From (30), we can derive equation (31):

e y t 1 | H n + 1 ( x | λ ) n + 1 = e y t 1 t | t { H n + 1 ( x | λ ) n + 1 } = e y t 1 t | H n ( x | λ ) .
(31)

By (11) and (31), we get

e y t 1 t | H n ( x | λ ) = e y t 1 | H n + 1 ( x | λ ) n + 1 = 1 n + 1 { H n + 1 ( y | λ ) H n + 1 ( λ ) } = 0 y H n ( u | λ ) d u .
(32)

Therefore, by (32), we obtain the following corollary.

Corollary 5 For n0, we have

e y t 1 t | H n ( x | λ ) = 0 y H n (u|λ)du.

Let P(λ)={p(x)Q(λ)[x]|degp(x)n} be a vector space over Q(λ).

For p(x) P n (λ), let us take

p(x)= k = 0 n b k H k (x|λ).
(33)

By Proposition 1, H n (x|λ) is an Appell sequence for g λ (t)= e t λ 1 λ where λ(1)C. Thus, we have

e t λ 1 λ t k | H n ( x | λ ) =n! δ n , k .
(34)

From (33) and (34), we can derive

e t λ 1 λ t k | p ( x ) = l = 0 n b l e t λ 1 λ t k | H l ( x | λ ) = l = 0 n b l l ! δ l , k = k ! b k .
(35)

Thus, by (35), we get

b k = 1 k ! e t λ 1 λ t k | p ( x ) = 1 k ! ( 1 λ ) ( e t λ ) t k | p ( x ) = 1 k ! ( 1 λ ) e t λ | p ( k ) ( x ) .
(36)

From (11) and (36), we have

b k = 1 k ! ( 1 λ ) { p ( k ) ( 1 ) λ p ( k ) ( 0 ) } ,
(37)

where p ( k ) (x)= d k p ( x ) d x k .

Therefore, by (37), we obtain the following theorem.

Theorem 6 For p(x) P n (λ), let us assume that p(x)= k = 0 n b k H k (x|λ). Then we have

b k = 1 k ! ( 1 λ ) { p ( k ) ( 1 ) λ p ( k ) ( 0 ) } ,

where p ( k ) (1)= d k p ( x ) d x k | x = 1 .

The higher-order Frobenius-Euler polynomials are defined by

( 1 λ e t λ ) r e x t = n = 0 H n ( r ) (x|λ) t n n ! ,
(38)

where λ(1)C and rN (see [4, 11]).

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. From (38), we have

H n ( r ) ( x ) = l = 0 n ( n l ) H n l ( r ) ( λ ) x l = n 1 + + n r = n ( n n 1 , , n r ) H n 1 ( x | λ ) H n r ( x | λ ) .
(39)

Note that H n ( r ) (x|λ) is a monic polynomial of degree n with coefficients in Q(λ).

For rN, λ(1)C, let g λ r (t)= ( e t λ 1 λ ) r . Then we easily see that g λ r (t) is an invertible series.

From (38) and (39), we have

1 g λ r ( t ) e x t = n = 0 H n ( r ) (x|λ) t n n ! ,
(40)

and

t H n ( r ) (x|λ)=n H n 1 ( r ) (x|λ).
(41)

By (40), we get

1 g λ r ( t ) x n = H n ( r ) (x|λ)(n Z + ,rN).
(42)

Therefore, by (41) and (42), we obtain the following proposition.

Proposition 7 For n Z + , H n ( r ) (x|λ) is an Appell sequence for

g λ r (t)= ( e t λ 1 λ ) r .

Moreover,

1 g λ r ( t ) x n = H n ( r ) (x|λ)andt H n ( r ) (x|λ)=n H n 1 ( r ) (x|λ).

Remark Note that

1 λ e t λ | x n = H n (λ).
(43)

From (43), we have

(44)
(45)

By (43), (44), and (45), we get

n = i 1 + + i r ( n i 1 , , i r ) H i 1 (λ) H i r (λ)= H n ( r ) (λ).

Let us take p(x) P n (λ) with

p(x)= k = 0 n C k ( r ) H k ( r ) (x|λ).
(46)

From the definition of Appell sequences, we have

( e t λ 1 λ ) r | H n ( r ) ( x | λ ) =n! δ n , k .
(47)

By (46) and (47), we get

( e t λ 1 λ ) r t k | p ( x ) = l = 0 n C l ( r ) ( e t λ 1 λ ) r t k | H l ( x | λ ) = l = 0 n C l ( r ) l ! δ l , k = k ! C k ( r ) .
(48)

Thus, from (48), we have

C k ( r ) = 1 k ! ( e t λ 1 λ ) r t k | p ( x ) = 1 k ! ( 1 λ ) r ( e t λ ) r t k | p ( x ) = 1 k ! ( 1 λ ) r l = 0 r ( r l ) ( λ ) r l e l t | p ( k ) ( x ) = 1 k ! ( 1 λ ) r l = 0 r ( r l ) ( λ ) r l p ( k ) ( l ) .
(49)

Therefore, by (46) and (49), we obtain the following theorem.

Theorem 8 For p(x) P n (λ), let

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

Then we have

C k ( r ) = 1 k ! ( 1 λ ) r l = 0 r ( r l ) ( λ ) r l p ( k ) (l),

where rN and p ( k ) (l)= d k p ( x ) d x k | x = l .

Remark Let S n (x) be a Sheffer sequence for (g(t),f(t)). Then Sheffer identity is given by

S n (x+y)= k = 0 n ( n k ) P k (y) S n k (x)= k = 0 n ( n k ) P k (x) S n k (y),
(50)

where P k (y)=g(t) S k (y) is associated to f(t) (see [1, 2]).

From (21), Proposition 1, and (50), we have

H n ( x + y | λ ) = k = 0 n ( n k ) P k ( y ) S n k ( x ) = k = 0 n ( n k ) H n k ( y | λ ) x k .

By Proposition 7 and (50), we get

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

Let α(0)C. Then we have

H n (αx|λ)= α n g λ ( t ) g λ ( t α ) H n (x|λ).

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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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Correspondence to Taekyun Kim.

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Kim, D.S., Kim, T. Some identities of Frobenius-Euler polynomials arising from umbral calculus. Adv Differ Equ 2012, 196 (2012). https://doi.org/10.1186/1687-1847-2012-196

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Keywords

  • Vector Space
  • Formal Power Series
  • Linear Functional
  • Monic Polynomial
  • Multiplicative Inverse
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