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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

$〈 L + M | p ( x ) 〉 = 〈 L | p ( x ) 〉 + 〈 M | p ( x ) 〉 , 〈 c L | p ( x ) 〉 = c 〈 L | p ( x ) 〉 ( see  ) ,$

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

(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

$p ( k ) (x)= d k p ( x ) d x k = ∑ l = k ∞ 〈 t l | p ( x ) 〉 l ! l(l−1)⋯(l−k+1) x l − k .$

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 $n,k≥0$. 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 ). 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

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

$1 g ( t ) x n = S n (x)if and only if x n =g(t) S n (x)(n≥0).$
(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 $λ(≠1)∈C$, $n≥0$, 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

$H k + 1 (x|λ)= ( x − g λ ′ ( t ) g λ ( t ) ) H k (x|λ)(k≥0).$

From (16), we have

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

By (25), we get

$H n (x+1|λ)−λ H n (x|λ)=(1−λ) x n .$
(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 $k≥0$, we have

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

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 $λ(≠1)∈C$, $n≥0$, 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 $n≥0$, 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 $r∈N$ (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 $r∈N$, $λ(≠1)∈C$, 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

$1 g λ r ( t ) x n = H n ( r ) (x|λ)(n∈ Z + ,r∈N).$
(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 $r∈N$ 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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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 