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Higher-order Euler-type polynomials and their applications

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.

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 nN.

N(M)=( N 1 , N 2 ,, N M ),

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

Let aN 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 0ka1, 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 PC[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 |<2π.

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 |<2π.
  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 kN and

|t+ j = 1 M 2 π i N j |<2π.

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 |<2π.

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 1uC. 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 |<2π

(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|<|lnλ|.

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 a1mod(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,NN (cf. [1]).

In this section, we give the following theorem.

Theorem 2.1 Let a, N 1 , N 2 ,, N M N and a1(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 a1(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 nN. 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 nN. 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.

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Acknowledgements

Dedicated to Professor Hari M Srivastava.

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

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Aygunes, A.A. Higher-order Euler-type polynomials and their applications. Fixed Point Theory Appl 2013, 40 (2013). https://doi.org/10.1186/1687-1812-2013-40

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