Higher-order Euler-type polynomials and their applications

R E S E A R C H

Open Access

Higher-order Euler-type polynomials and

their applications

Aykut Ahmet Aygunes

*_{Correspondence:} [email protected] Department of Mathematics, Faculty of Science, University of Akdeniz, Antalya, TR-07058, Turkey

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 nota-tion 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={, , . . .},N={, , , . . .}=N∪ {}. Also, as usual,Zdenotes the set of integers,R denotes the set of real numbers andCdenotes the set of complex numbers. We assume thatln(z) denotes the principal branch of the multi-valued functionln(z) with an imag-inary part(ln(z)) constrained by –π <(ln(z))≤π. Furthermore, n=  ifn= , and n_{=  ifn∈N.}

N(M) = (N,N, . . . ,NM),

whereM∈NandN,N, . . . ,NM∈N.

Leta∈Nandχa,N(M)be a function depending ona,N,N, . . . ,NMsuch that

χa,N(M):N→C.

χa,N(M)is defined by

χa,N(M)(k) = M

j=

ξk(Nj),

where ≤k≤a– ,j∈ {, , . . . ,M}and

ξ(Nj) =e

πi Nj_.

χa,N(M)satisfies the following properties:

(i) χa,N(M)is a periodic function withNN· · ·NM.

(ii) If we takeN≥andN=N=· · ·=NM= , we have

χa,(N,,,...,)(k) =ξ

k_(N

)ξk()ξk()· · ·ξk() =ξk(N).

We note that replacingN(M) by (N, , , . . . , ),χa,N(M)is reduced toξk(N) (cf.[]).

LetC[x] be a ring of polynomials with complex coefficients. By usingχa,N(M), we give

the following definition.

Definition .[] LetP∈C[x]. The generalized partial Hecke operator ofTχa,N(M) is de-fined by

Tχa,N(M)

P(x)=

a–

k=

χa,N(M)(k)P

x+k a

.

The operatorTχ_a,N(M)satisfies the following properties:

(i) Tχa,N(M)is linear onC[x]and

Tχa,N(M):C[x]→C[x].

(ii) Tχa,N(M)preserves the degree of the polynomials onC[x].

(iii) If we takeN≥andN=N=· · ·=NM= , we have

Tχa,N_

P(x)=

a–

k=

ξk(N)P

x+k a

.

Remark . SettingN(M) = (N, , , . . . , ),Tχa,(N,,,...,)is reduced toTχa,N(cf.[]).

The generating function of generalized Euler-type numbersPn,N(M)is given by

FN(M)(t) =

∞

n=

Pn,N(M)

tn

n!=

M

j=ξ(Nj) – 

– +etM j=ξ(Nj)

[].

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

Definition .[] The polynomialPn,N(M)is defined by means of the following generating

function:

FN(M)(t,x) =

∞

n=

Pn,N(M)(x)

tn

n!=

((M_j=ξ(Nj)) – )etx

(M_j=ξ(Nj))et– 

where

t+

M

j=

πi Nj

< π.

The polynomialPn,N(M)satisfies the following properties:

(i) Pn,N(M)∈C[x].

(ii) Pn,N(M)is a polynomial with degreenand depends onN,N, . . . ,NM.

(iii) If we takeN≥andN=N=· · ·=NM= , we have

∞

n=

Pn,N(x) tn n!=

(ξN– )e

tx

ξNet–  ,

where

t+πi

N < π.

(iv) We derive the following functional equation:

FN(M)(t,x) =FN(M)(t)etx, ()

so that, obviously,

Pn,N(M)() =Pn,N(M).

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

Definition . Euler-type numbers with orderk, P_n,N(k)_(M), are defined by means of the following generating functions:

F(k) N(M)(t) =

∞

n=

P(k)_n,N(M)t

n

n!, ()

wherek∈Nand

t+

M

j=

πi Nj

< π.

Euler-type polynomials with orderkare given by the following functional equation:

F(k)

N(M)(t,x) =F (k) N(M)(t)e

tx₌

∞

n=

P(k)_n,N(M)(x)t

n

n!. ()

We see that

F()

Thus we obtain

P()_n,N(M)(x) =xn.

Remark . Substitutingk=  into (), we get (). Therefore, () reduces to (); that is,

P()_n,N(M)(x) =Pn,N(M)(x)

so that, obviously,

P()_n,N(M)() =Pn,N(M).

By using () and (), we obtain

∞

n=

P_n,N(k)_(M)(x)t

n

n! = ∞

n=

P_n,N(k)_(M)t

n

n! ∞

n=

xnt

n

n!.

Therefore, we get the following theorem.

Theorem .

P(k)_n,N(M)(x) =

n

j=

n j

xn–jP(k)_j,N(M). ()

Hence, we arrive at the following definition.

Definition . Euler-type polynomials with orderk,P_n,N(k)_(M), are defined by means of the following generating functions:

F(k)

N(M)(t,x) =

∞

n=

P(k)_n,N(M)(x)t

n

n!, ()

where

t+

M

j=

πi Nj

< π.

Note that there is one generating function for each value ofk. These are given explicitly as follows:

F(k)

N(M)(t,x) =

_{– +}M j=ξ(Nj)

– +etM j=ξ(Nj)

k

etx

= ∞

n=

P(k)_n,N(M)(x)t

n

n!.

We derive the following functional equation:

F(k+l)

N(M)(t,x) =F (k)

N(M)(t,x)F (l)

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

Theorem .

P(k+l)_n,N(M)(x) =

n

j=

n j

P(k)_j,N(M)(x)P(l)_n–j,N(M). ()

Proof By using (), () and (), we get

∞

n=

P_n,N(k+l)_(M)(x)t

n

n! = ∞

n= ⎛ ⎝n

j=

n j

P(k)_j,N(M)(x)P_n–j,N(M)(l)

⎞ ⎠tn

n!.

By comparing the coefficients oft_n!non both sides of the above equation, we get the desired

result.

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

Corollary .

P(k+l)_n,N(M)=

n

j=

n j

P(k)_j,N(M)P_n–j,N(l) _(M).

By differentiating both sides of equation () with respect to the variablex, we obtain the following higher-order differential equation:

∂j

∂xjFN(M)(t,x) =t j_F

N(M)(t,x). ()

Remark . SettingN(M) = (N, , , . . . , ),Pn,(N,,,...,)is reducedPn,N(x) (cf.[]). There-forePn,N(x) was defined by generalized Bernoulli-Euler polynomials in [] as follows:

∞

n=

Pn,N(x)

tn

n!=

⎧ ⎨ ⎩

tetx

et_–, N= ,

(ξN–)etx

ξNet– , N≥,

so that, obviously,

Pn,(x) =Bn(x)

and

Pn,(x) =En(x).

HereBn(x) andEn(x) are Bernoulli polynomials and Euler polynomials, respectively (cf.

[–]).

Letube an algebraic number such that =u∈C. Then the Frobenius-Euler polynomial

Hn(x,u) is defined by

 –u et_–ue

tx₌

∞

n=

Hn(x,u)

tn

n!,

where

t+ln

u

< π

(cf.[–]).

Remark . Frobenius-Euler number is denoted byHn(u) such thatHn(,u) =Hn(u).

Also,Hn(x, –) =En(x) (cf.[–]).

By using Frobenius-Euler numbers, one can obtain the Frobenius-Euler polynomials as follows:

Hn(x,u) = n

j=

n j

xn–jHj(u)

(cf.[–]).

The Apostol-Bernoulli polynomial is defined as follows.

Definition .[, ] The Apostol-Bernoulli polynomialBn(x,λ) is defined by

t

λet_{– }e tx₌

∞

n= Bn(x,λ)

tn

n!,

whereλis the arbitrary real or complex parameter and |t|<|lnλ|.

Remark . Forλ= , we obtain thatBn(x, ) =Bn(x) (cf.[–]). 2 A functional equation of generalized Euler-type polynomials

Bayad, Aygunes and Simsek showed that fora≡mod(N), there exists a unique sequence of monic polynomials (Pn,N)n∈NinQ(ξN)[x] withdegPn,N=nsuch that

Tχa,N

Pn,N(x)

=a–nPn,N(x),

wherea,N∈N(cf.[]).

In this section, we give the following theorem.

Theorem . Let a,N,N, . . . ,NM∈Nand a≡(modNN· · ·NM).Then there exists a

sequence(Pn,N(M))n∈Nin

Qξ(N)ξ(N)· · ·ξ(NM)

with

degPn,N(M)=n

such that

Tχa,N(M)

Pn,N(M)(x)

=a–nPn,N(M)(x). ()

Proof SincePn,N(M)∈C[x] andTχa,N(M):C[x]→C[x], we get

Tχa,N(M)

Pn,N(M)(x)

=

a–

k=

χa,N(M)(k)Pn,N(M)

x+k a

.

From the definition ofχa,N(M)(k), we have

Tχa,N(M)

Pn,N(M)(x) = a– k= _M j= e πik

Nj

Pn,N(M)

x+k a

.

By using the generating function ofPn,N(M)(x), we get

∞ n= a– k= _M j= e πik

Nj

Pn,N(M)

x+k a tn n! = a– k= _M j= e πik

Nj

_∞

n=

Pn,N(M)

x+k a tn n! = a– k= _M j= e πik

Nj

((M_j=e πi

Nj_{) – )e}t(x+ka )

(M_j=e πi

Nj_)et_{– }

=((

M j=e

πi Nj_{) – )e}txa

(M_j=e πi

Nj_)et_{– } a– k= exp _M j= e πik

Nj exp tk a =(( M j=e

πi Nj_{) – )e}txa

(M_j=e πi

Nj_)et_{– } a– k= exp t a+ M j=

πi Nj

k

=((

M j=e

πi Nj_{) – )e}txa

(M_j=e πi

Nj_)et_{– }

et_(exp(M j=

πi Nj))

a_{– }

eta_(exp(M

j=Nπji)) –  .

Sincea≡(modNN· · ·NM), the following relation holds:

exp

_M

j=

πi Nj a =exp _M j=

πi Nj = M j= e πi

Nj_.

Therefore, we have

∞ n= _a– k= _M j= e πik

Nj

Pn,N(M)

x+k a

tn

n!= ∞

n=

a–nPn,N(M)(x)

tn

By comparing the coefficients of t_n!n on both sides of the above equation, we get the

de-sired result.

Remark . A different proof of () is given in []. If we takeN≥ andN=N=· · ·=

NM= , we have the following functional equation:

Tχa,N

Pn,N(x)

=a–nPn,N(x)

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

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 . Let n∈N.Then we have

Pn+,N(M)(x) =Pn,N(M)(x) + M

j=ξ(Nj)

 –M_j=ξ(Nj) n

k=

n k

P()_k,N(M)(x).

Proof By differentiating both sides of equation () with respect to the variablet, we have

∞

n=

Pn+,N(M)(x)

tn

n! =

∂

∂tFN(M)(t,x)

=FN(M)(t,x) + M

j=ξ(Nj)

 –M_j=ξ(Nj)

etetxFN(M)(t) 

= ∞

n=

Pn,N(M)(x)

tn

n!+

M j=ξ(Nj)

 –M_j=ξ(Nj)

et

_∞

n=

P()_n,N(M)(x)t

n

n!

.

Therefore, we obtain

∞

n=

Pn+,N(M)(x)

tn

n!= ∞

n=

Pn,N(M)(x) + M

j=ξ(Nj)

 –M_j=ξ(Nj) n

k=

n k

P()_k,N(M)(x)

tn

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 Pn,N(M)(x) and

Frobenius-Euler polynomials.

Theorem .[] Let n∈N.Then we have

Pn,N(M)(x) =Hn

x,

M

j=



ξ(Nj)

Proof By using the generating function ofPn,N(M)(x), we have

∞

n=

Pn,N(M)(x)

tn n! =

∞

n=

Hn

x,

M

j=



ξ(Nj)

tn 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 betweenPn,N(M)(x) and Apostol-Bernoulli

polynomials.

Theorem .[] Let n∈N.Then we have

Pn–,N(M)(x) = _M

j=

ξ(Nj) – 



nBn

x,

M

j=

ξ(Nj)

.

Proof We arrange the generating function of generalized Euler-type polynomials as fol-lows:

∞

n=

Pn–,N(M)

tn–

(n– )!=

M

j=ξ(Nj) – 

etM

j=ξ(Nj) – 

ext.

Therefore, we have

∞

n=

Pn–,N(M)

tn–

(n– )!= ∞

n=



n

_M

j=

ξ(Nj) – 

Bn

x,

M

j=

ξ(Nj)

tn–

(n– )!.

In the above equation, if we compare the coefficients of _(n–)!tn– , we get the desired result.

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

Theorem . Let n∈N.Then we have

Pn,N(M)(x) = n

j=

n j

xn–jPj,N(M). ()

Proof of () is the same as that of (), so we omit it [].

P,N(M)=



χ_a,N(M)– – ,

P,N(M)=

 (χ_a,N–_(M)– ) +



χ_a,N(M)– – ,

P,N(M)=

 (χ_a,N–_(M)– ) +

 (χ_a,N(M)– – )+



and

P,N(M)=

 (χ_a,N(M)– – )+

 (χ_a,N–_(M)– ) +

 (χ_a,N–_(M)– ) +



χ_a,N(M)– – .

By using (), we have the following list for the generalized Euler-type polynomials:

P,N(M)(x) = ,

P,N(M)(x) =x+



χ_a,N(M)– – ,

P,N(M)(x) =x+x



χ_a,N(M)– – 

+

 (χ_a,N(M)– – )+



χ_a,N–_(M)– 

,

P,N(M)(x) =x+x



χ_a,N(M)– – 

+x

 (χ_a,N(M)– – ) +



χ_a,N–_(M)– 

+

 (χ–

a,N(M)– )

+ 

(χ–

a,N(M)– )

+ 

χ– a,N(M)– 

and

P,N(M)(x) =x+x



χ_a,N–_(M)– 

+x

 (χ_a,N(M)– – )+



χ_a,N–_(M)– 

+x

 (χ_a,N(M)– – )+

 (χ_a,N(M)– – ) +



χ_a,N–_(M)– 

+

 (χ–

a,N(M)– )

+ 

(χ–

a,N(M)– )

+ 

(χ–

a,N(M)– )

+ 

χ– a,N(M)– 

.

Competing interests

The author declares that he has no competing interests.

Author’s contributions

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

Acknowledgements

Dedicated to Professor Hari M Srivastava.

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

Received: 11 December 2012 Accepted: 6 February 2013 Published: 26 February 2013

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doi:10.1186/1687-1812-2013-40