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Higher-order Euler-type polynomials and their applications

Fixed Point Theory and Applications20132013:40

https://doi.org/10.1186/1687-1812-2013-40

  • Received: 11 December 2012
  • Accepted: 6 February 2013
  • Published:

Abstract

In this paper, we construct generating functions for higher-order Euler-type polynomials and numbers. By using the generating functions, we obtain functional equations related to a generalized partial Hecke operator and Euler-type polynomials and numbers. A special case of higher-order Euler-type polynomials is eigenfunctions for the generalized partial Hecke operators. Moreover, we give not only some properties, but also applications for these polynomials and numbers.

AMS Subject Classification:08A40, 11F25, 11F60, 11B68, 30D05.

Keywords

  • generalized partial Hecke operators
  • higher-order Euler-type polynomials
  • higher-order Euler-type numbers
  • Apostol-Bernoulli polynomials
  • Frobenius-Euler polynomials
  • Euler polynomials
  • Euler numbers
  • functional equation
  • generating functions

1 Introduction

In this section, we define generalized partial Hecke operators and we give some notation for these operators. Also, we define generalized Euler-type polynomials, Apostol-Bernoulli polynomials and Frobenius-Euler polynomials.

Throughout this paper, we use the following notations:

N = { 1 , 2 , } , N 0 = { 0 , 1 , 2 , } = N { 0 } . Also, as usual, denotes the set of integers, denotes the set of real numbers and denotes the set of complex numbers. We assume that ln ( z ) denotes the principal branch of the multi-valued function ln ( z ) with an imaginary part ( ln ( z ) ) constrained by π < ( ln ( z ) ) π . Furthermore, 0 n = 1 if n = 0 , and 0 n = 0 if n N .
N ( M ) = ( N 1 , N 2 , , N M ) ,

where M N and N 1 , N 2 , , N M N .

Let a N and χ a , N ( M ) be a function depending on a , N 1 , N 2 , , N M such that
χ a , N ( M ) : N 0 C .
χ a , N ( M ) is defined by
χ a , N ( M ) ( k ) = j = 1 M ξ k ( N j ) ,
where 0 k a 1 , j { 1 , 2 , , M } and
ξ ( N j ) = e 2 π i N j .
χ a , N ( M ) satisfies the following properties:
  1. (i)

    χ a , N ( M ) is a periodic function with N 1 N 2 N M .

     
  2. (ii)
    If we take N 1 2 and N 2 = N 3 = = N M = 1 , we have
    χ a , ( N 1 , 1 , 1 , , 1 ) ( k ) = ξ k ( N 1 ) ξ k ( 1 ) ξ k ( 1 ) ξ k ( 1 ) = ξ k ( N 1 ) .
     

We note that replacing N ( M ) by ( N 1 , 1 , 1 , , 1 ) , χ a , N ( M ) is reduced to ξ k ( N 1 ) (cf. [1]).

Let C [ x ] be a ring of polynomials with complex coefficients. By using χ a , N ( M ) , we give the following definition.

Definition 1.1 [2]

Let P C [ x ] . The generalized partial Hecke operator of T χ a , N ( M ) is defined by
T χ a , N ( M ) ( P ( x ) ) = k = 0 a 1 χ a , N ( M ) ( k ) P ( x + k a ) .
The operator T χ a , N ( M ) satisfies the following properties:
  1. (i)
    T χ a , N ( M ) is linear on C [ x ] and
    T χ a , N ( M ) : C [ x ] C [ x ] .
     
  2. (ii)

    T χ a , N ( M ) preserves the degree of the polynomials on C [ x ] .

     
  3. (iii)
    If we take N 1 2 and N 2 = N 3 = = N M = 1 , we have
    T χ a , N 1 ( P ( x ) ) = k = 0 a 1 ξ k ( N 1 ) P ( x + k a ) .
     

Remark 1.2 Setting N ( M ) = ( N 1 , 1 , 1 , , 1 ) , T χ a , ( N 1 , 1 , 1 , , 1 ) is reduced to T χ a , N 1 (cf. [1]).

The generating function of generalized Euler-type numbers P n , N ( M ) is given by
F N ( M ) ( t ) = n = 0 P n , N ( M ) t n n ! = j = 1 M ξ ( N j ) 1 1 + e t j = 1 M ξ ( N j )

[2].

Now, we give the definition of Euler-type polynomials as follows.

Definition 1.3 [2]

The polynomial P n , N ( M ) is defined by means of the following generating function:
F N ( M ) ( t , x ) = n = 0 P n , N ( M ) ( x ) t n n ! = ( ( j = 1 M ξ ( N j ) ) 1 ) e t x ( j = 1 M ξ ( N j ) ) e t 1 ,
(1)
where
| t + j = 1 M 2 π i N j | < 2 π .
The polynomial P n , N ( M ) satisfies the following properties:
  1. (i)

    P n , N ( M ) C [ x ] .

     
  2. (ii)

    P n , N ( M ) is a polynomial with degree n and depends on N 1 , N 2 , , N M .

     
  3. (iii)
    If we take N 1 2 and N 2 = N 3 = = N M = 1 , we have
    n = 0 P n , N 1 ( x ) t n n ! = ( ξ N 1 1 ) e t x ξ N 1 e t 1 ,
     
where
| t + 2 π i N 1 | < 2 π .
  1. (iv)
    We derive the following functional equation:
    F N ( M ) ( t , x ) = F N ( M ) ( t ) e t x ,
    (2)
    so that, obviously,
    P n , N ( M ) ( 0 ) = P n , N ( M ) .
     

We now are ready to define Euler-type numbers and polynomials with order k.

Definition 1.4 Euler-type numbers with order k, P n , N ( M ) ( k ) , are defined by means of the following generating functions:
F N ( M ) ( k ) ( t ) = n = 0 P n , N ( M ) ( k ) t n n ! ,
(3)
where k N and
| t + j = 1 M 2 π i N j | < 2 π .
Euler-type polynomials with order k are given by the following functional equation:
F N ( M ) ( k ) ( t , x ) = F N ( M ) ( k ) ( t ) e t x = n = 0 P n , N ( M ) ( k ) ( x ) t n n ! .
(4)
We see that
F N ( M ) ( 0 ) ( t , x ) = e t x .
Thus we obtain
P n , N ( M ) ( 0 ) ( x ) = x n .
Remark 1.5 Substituting k = 1 into (4), we get (2). Therefore, (3) reduces to (1); that is,
P n , N ( M ) ( 1 ) ( x ) = P n , N ( M ) ( x )
so that, obviously,
P n , N ( M ) ( 1 ) ( 0 ) = P n , N ( M ) .
By using (4) and (3), we obtain
n = 0 P n , N ( M ) ( k ) ( x ) t n n ! = n = 0 P n , N ( M ) ( k ) t n n ! n = 0 x n t n n ! .

Therefore, we get the following theorem.

Theorem 1.6
P n , N ( M ) ( k ) ( x ) = j = 0 n ( n j ) x n j P j , N ( M ) ( k ) .
(5)

Hence, we arrive at the following definition.

Definition 1.7 Euler-type polynomials with order k, P n , N ( M ) ( k ) , are defined by means of the following generating functions:
F N ( M ) ( k ) ( t , x ) = n = 0 P n , N ( M ) ( k ) ( x ) t n n ! ,
(6)
where
| t + j = 1 M 2 π i N j | < 2 π .
Note that there is one generating function for each value of k. These are given explicitly as follows:
F N ( M ) ( k ) ( t , x ) = ( 1 + j = 1 M ξ ( N j ) 1 + e t j = 1 M ξ ( N j ) ) k e t x = n = 0 P n , N ( M ) ( k ) ( x ) t n n ! .
We derive the following functional equation:
F N ( M ) ( k + l ) ( t , x ) = F N ( M ) ( k ) ( t , x ) F N ( M ) ( l ) ( t ) .
(7)

By using the above functional equation, we arrive at the following theorem.

Theorem 1.8
P n , N ( M ) ( k + l ) ( x ) = j = 0 n ( n j ) P j , N ( M ) ( k ) ( x ) P n j , N ( M ) ( l ) .
(8)
Proof By using (3), (6) and (7), we get
n = 0 P n , N ( M ) ( k + l ) ( x ) t n n ! = n = 0 ( j = 0 n ( n j ) P j , N ( M ) ( k ) ( x ) P n j , N ( M ) ( l ) ) t n n ! .

By comparing the coefficients of t n n ! on both sides of the above equation, we get the desired result. □

Substituting x = 0 into (8), we obtain a convolution formula for the numbers by the following corollary.

Corollary 1.9
P n , N ( M ) ( k + l ) = j = 0 n ( n j ) P j , N ( M ) ( k ) P n j , N ( M ) ( l ) .
By differentiating both sides of equation (2) with respect to the variable x, we obtain the following higher-order differential equation:
j x j F N ( M ) ( t , x ) = t j F N ( M ) ( t , x ) .
(9)
Remark 1.10 Setting N ( M ) = ( N 1 , 1 , 1 , , 1 ) , P n , ( N 1 , 1 , 1 , , 1 ) is reduced P n , N 1 ( x ) (cf. [1]). Therefore P n , N ( x ) was defined by generalized Bernoulli-Euler polynomials in [1] as follows:
n = 0 P n , N ( x ) t n n ! = { t e t x e t 1 , N = 1 , ( ξ N 1 ) e t x ξ N e t 1 , N 2 ,
so that, obviously,
P n , 1 ( x ) = B n ( x )
and
P n , 2 ( x ) = E n ( x ) .

Here B n ( x ) and E n ( x ) are Bernoulli polynomials and Euler polynomials, respectively (cf. [119]).

The Frobenius-Euler polynomial is defined as follows:

Let u be an algebraic number such that 1 u C . Then the Frobenius-Euler polynomial H n ( x , u ) is defined by
1 u e t u e t x = n = 0 H n ( x , u ) t n n ! ,
where
| t + ln 1 u | < 2 π

(cf. [119]).

Remark 1.11 Frobenius-Euler number is denoted by H n ( u ) such that H n ( 0 , u ) = H n ( u ) . Also, H n ( x , 1 ) = E n ( x ) (cf. [119]).

By using Frobenius-Euler numbers, one can obtain the Frobenius-Euler polynomials as follows:
H n ( x , u ) = j = 0 n ( n j ) x n j H j ( u )

(cf. [119]).

The Apostol-Bernoulli polynomial is defined as follows.

Definition 1.12 [3, 16]

The Apostol-Bernoulli polynomial B n ( x , λ ) is defined by
t λ e t 1 e t x = n = 0 B n ( x , λ ) t n n ! ,
where λ is the arbitrary real or complex parameter and
| t | < | ln λ | .

Remark 1.13 For λ = 1 , we obtain that B n ( x , 1 ) = B n ( x ) (cf. [119]).

2 A functional equation of generalized Euler-type polynomials

Bayad, Aygunes and Simsek showed that for a 1 mod ( N ) , there exists a unique sequence of monic polynomials ( P n , N ) n N 0 in Q ( ξ N ) [ x ] with deg P n , N = n such that
T χ a , N ( P n , N ( x ) ) = a n P n , N ( x ) ,

where a , N N (cf. [1]).

In this section, we give the following theorem.

Theorem 2.1 Let a , N 1 , N 2 , , N M N and a 1 ( mod N 1 N 2 N M ) . Then there exists a sequence ( P n , N ( M ) ) n N 0 in
Q ( ξ ( N 1 ) ξ ( N 2 ) ξ ( N M ) ) [ x ]
with
deg P n , N ( M ) = n
such that
T χ a , N ( M ) ( P n , N ( M ) ( x ) ) = a n P n , N ( M ) ( x ) .
(10)
Proof Since P n , N ( M ) C [ x ] and T χ a , N ( M ) : C [ x ] C [ x ] , we get
T χ a , N ( M ) ( P n , N ( M ) ( x ) ) = k = 0 a 1 χ a , N ( M ) ( k ) P n , N ( M ) ( x + k a ) .
From the definition of χ a , N ( M ) ( k ) , we have
T χ a , N ( M ) ( P n , N ( M ) ( x ) ) = k = 0 a 1 ( j = 1 M e 2 π i k N j ) P n , N ( M ) ( x + k a ) .
By using the generating function of P n , N ( M ) ( x ) , we get
n = 0 k = 0 a 1 ( j = 1 M e 2 π i k N j ) P n , N ( M ) ( x + k a ) t n n ! = k = 0 a 1 ( j = 1 M e 2 π i k N j ) n = 0 P n , N ( M ) ( x + k a ) t n n ! = k = 0 a 1 ( j = 1 M e 2 π i k N j ) ( ( j = 1 M e 2 π i N j ) 1 ) e t ( x + k a ) ( j = 1 M e 2 π i N j ) e t 1 = ( ( j = 1 M e 2 π i N j ) 1 ) e t x a ( j = 1 M e 2 π i N j ) e t 1 k = 0 a 1 ( exp ( j = 1 M e 2 π i k N j ) ) exp ( t k a ) = ( ( j = 1 M e 2 π i N j ) 1 ) e t x a ( j = 1 M e 2 π i N j ) e t 1 k = 0 a 1 ( exp ( t a + j = 1 M 2 π i N j ) ) k = ( ( j = 1 M e 2 π i N j ) 1 ) e t x a ( j = 1 M e 2 π i N j ) e t 1 e t ( exp ( j = 1 M 2 π i N j ) ) a 1 e t a ( exp ( j = 1 M 2 π i N j ) ) 1 .
Since a 1 ( mod N 1 N 2 N M ) , the following relation holds:
( exp ( j = 1 M 2 π i N j ) ) a = exp ( j = 1 M 2 π i N j ) = j = 1 M e 2 π i N j .
Therefore, we have
n = 0 ( k = 0 a 1 ( j = 1 M e 2 π i k N j ) P n , N ( M ) ( x + k a ) ) t n n ! = n = 0 a n P n , N ( M ) ( x ) t n n ! .

By comparing the coefficients of t n n ! on both sides of the above equation, we get the desired result. □

Remark 2.2 A different proof of (10) is given in [2]. If we take N 1 2 and N 2 = N 3 = = N M = 1 , we have the following functional equation:
T χ a , N 1 ( P n , N 1 ( x ) ) = a n P n , N 1 ( x )

which is satisfied for generalized Bernoulli-Euler polynomials in [1].

3 Some properties of generalized Euler-type polynomials

In this section, we obtain some relations between generalized Euler-type polynomials, Apostol-Bernoulli polynomials and Frobenius-Euler polynomials. Also, we give a formula to obtain the generalized Euler-type polynomials.

Theorem 3.1 Let n N . Then we have
P n + 1 , N ( M ) ( x ) = P n , N ( M ) ( x ) + j = 1 M ξ ( N j ) 1 j = 1 M ξ ( N j ) k = 0 n ( n k ) P k , N ( M ) ( 2 ) ( x ) .
Proof By differentiating both sides of equation (2) with respect to the variable t, we have
n = 0 P n + 1 , N ( M ) ( x ) t n n ! = t F N ( M ) ( t , x ) = F N ( M ) ( t , x ) + ( j = 1 M ξ ( N j ) 1 j = 1 M ξ ( N j ) ) e t e t x ( F N ( M ) ( t ) ) 2 = n = 0 P n , N ( M ) ( x ) t n n ! + ( j = 1 M ξ ( N j ) 1 j = 1 M ξ ( N j ) ) e t ( n = 0 P n , N ( M ) ( 2 ) ( x ) t n n ! ) .
Therefore, we obtain
n = 0 P n + 1 , N ( M ) ( x ) t n n ! = n = 0 ( P n , N ( M ) ( x ) + j = 1 M ξ ( N j ) 1 j = 1 M ξ ( N j ) k = 0 n ( n k ) P k , N ( M ) ( 2 ) ( x ) ) t n n ! .

By comparing the coefficients of t n n ! , we obtain the desired result. □

In the following theorem, we give a relation between the polynomials P n , N ( M ) ( x ) and Frobenius-Euler polynomials.

Theorem 3.2 [2]

Let n N 0 . Then we have
P n , N ( M ) ( x ) = H n ( x , j = 1 M 1 ξ ( N j ) ) .
Proof By using the generating function of P n , N ( M ) ( x ) , we have
n = 0 P n , N ( M ) ( x ) t n n ! = n = 0 H n ( x , j = 1 M 1 ξ ( N j ) ) t n n ! .

In the above equation, if we compare the coefficients of t n n ! , we get the desired result. □

In the following theorem, we give a relation between P n , N ( M ) ( x ) and Apostol-Bernoulli polynomials.

Theorem 3.3 [2]

Let n N . Then we have
P n 1 , N ( M ) ( x ) = ( j = 1 M ξ ( N j ) 1 ) 1 n B n ( x , j = 1 M ξ ( N j ) ) .

Proof

We arrange the generating function of generalized Euler-type polynomials as follows:
n = 1 P n 1 , N ( M ) t n 1 ( n 1 ) ! = j = 1 M ξ ( N j ) 1 e t j = 1 M ξ ( N j ) 1 e x t .
Therefore, we have
n = 1 P n 1 , N ( M ) t n 1 ( n 1 ) ! = n = 1 ( 1 n ( j = 1 M ξ ( N j ) 1 ) B n ( x , j = 1 M ξ ( N j ) ) ) t n 1 ( n 1 ) ! .

In the above equation, if we compare the coefficients of t n 1 ( n 1 ) ! , we get the desired result. □

In the following theorem, it is possible to find the generalized Euler-type polynomials.

Theorem 3.4 Let n N 0 . Then we have
P n , N ( M ) ( x ) = j = 0 n ( n j ) x n j P j , N ( M ) .
(11)
Proof of (11) is the same as that of (5), so we omit it [2].
P 1 , N ( M ) = 1 χ a , N ( M ) 1 1 , P 2 , N ( M ) = 2 ( χ a , N ( M ) 1 1 ) 2 + 1 χ a , N ( M ) 1 1 , P 3 , N ( M ) = 6 ( χ a , N ( M ) 1 1 ) 3 + 6 ( χ a , N ( M ) 1 1 ) 2 + 1 χ a , N ( M ) 1 1
and
P 4 , N ( M ) = 24 ( χ a , N ( M ) 1 1 ) 4 + 36 ( χ a , N ( M ) 1 1 ) 3 + 14 ( χ a , N ( M ) 1 1 ) 2 + 1 χ a , N ( M ) 1 1 .
By using (11), we have the following list for the generalized Euler-type polynomials:
P 0 , N ( M ) ( x ) = 1 , P 1 , N ( M ) ( x ) = x + 1 χ a , N ( M ) 1 1 , P 2 , N ( M ) ( x ) = x 2 + x ( 2 χ a , N ( M ) 1 1 ) + ( 2 ( χ a , N ( M ) 1 1 ) 2 + 1 χ a , N ( M ) 1 1 ) , P 3 , N ( M ) ( x ) = x 3 + x 2 ( 3 χ a , N ( M ) 1 1 ) + x ( 6 ( χ a , N ( M ) 1 1 ) 2 + 3 χ a , N ( M ) 1 1 ) + ( 6 ( χ a , N ( M ) 1 1 ) 3 + 6 ( χ a , N ( M ) 1 1 ) 2 + 1 χ a , N ( M ) 1 1 )
and
P 4 , N ( M ) ( x ) = x 4 + x 3 ( 4 χ a , N ( M ) 1 1 ) + x 2 ( 12 ( χ a , N ( M ) 1 1 ) 2 + 6 χ a , N ( M ) 1 1 ) + x ( 24 ( χ a , N ( M ) 1 1 ) 3 + 24 ( χ a , N ( M ) 1 1 ) 2 + 4 χ a , N ( M ) 1 1 ) + ( 24 ( χ a , N ( M ) 1 1 ) 4 + 36 ( χ a , N ( M ) 1 1 ) 3 + 14 ( χ a , N ( M ) 1 1 ) 2 + 1 χ a , N ( M ) 1 1 ) .

Author’s contributions

The author completed the paper himself. The author read and approved the final manuscript.

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

The present investigation was supported by the Scientific Research Project Administration of Akdeniz University.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, University of Akdeniz, Antalya, TR-07058, Turkey

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