Generating function for q-Eulerian polynomials and their decomposition and applications

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R E S E A R C H

Open Access

Generating function for

q

-Eulerian

polynomials and their decomposition and

applications

Mustafa Alkan

*

_{and Yilmaz Simsek}

*_{Correspondence:}

[email protected] Department of Mathematics, Faculty of Science, Akdeniz University, Campus, Antalya, 07058, Turkey

Abstract

The aim of this paper is to deﬁne a generating function forq-Eulerian polynomials and numbers attached to any character

χ

of the finite cyclic groupG. We derive many functional equations,q-difference equations and partial deferential equations related to these generating functions. By using these equations, we find many properties ofq-Eulerian polynomials and numbers. Using the generating element of the finite cyclic groupGand the generating element of the subgroups ofG, we show that the generating function with a conductorfcan be written as a sum of the generating function with conductors which are less thanf.

MSC: 05A40; 11B83; 11B68; 11S80

Keywords: Euler numbers; Frobenius-Euler numbers; Frobenius-Euler polynomials; q-Frobenius-Euler polynomials;q-series; generating function; character

χ

of the ﬁnite abelian groupsG

1 Introduction

The theory of the family of Eulerian polynomials and numbers (Frobenius-Euler polyno-mials and numbers) has become a very important part of analytic number theory and other sciences, for example, engineering, computer, geometric design and mathematical physics. Euler numbers are related to the Brouwer fixed point theorem and vector fields. Therefore,q-Eulerian type numbers may be related to Brouwer fixed point theorem and vector fields [].

Recently, many diﬀerent special functions have been used to deﬁne and investigate

q-Eulerian numbers and polynomials, see details [–]. Therefore, we construct and in-vestigate various properties ofq-Eulerian numbers and polynomials, which are related to the many known polynomials and numbers such as Frobenius-Euler polynomials and numbers, Apostol-Euler polynomials and numbers, Euler polynomials and numbers.

Recently in [–], the authors defined a relationship between algebra and analysis. In de-tail, they made a new approximation between the subgroup and monoid presentation and special generating functions (such as Stirling numbers, Array polynomialsetc.). In this pa-per, since our priority aim is to define special numbers and polynomials, it is worth depict-ing these references as well. Then in this paper, applydepict-ing any group characterχof the finite cyclic groupGto a special generating function (which has been defined in []), we give a generalization ofq-Eulerian polynomials and numbers (q-Apostol-type Frobenius-Euler

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polynomials and numbers) and investigate their properties and some useful functional equations. Using a generating element of the subgroups ofGand a generating element of

G, we also decompose our generating function attached to the characters ofG, and so we obtain a new decomposition ofq-Eulerian numbers and polynomials attached to the char-acters of a subgroup ofGand the Dirichlet character ofG. This decomposition enables us to computeq-Apostol-type Frobenius-Euler polynomials and numbers more easily.

Throughout this paper, we assume thatq∈C, the set of complex numbers, with|q|< 

[x] = [x:q] = ⎧ ⎨ ⎩

–qx

–q ifq= ,

x ifq= .

We use the following standard notions.

N={, , . . .},N={, , , . . .}=N∪ {} and also, as usual,Rdenotes the set of real numbers,R+denotes the set of positive real numbers andCdenotes the set of complex numbers.

1.1 Characters of a groupG

We recall the deﬁnition and some properties of the character of an arbitrary group (G,·) (see []).

A non-zero complex-valued functionχdeﬁned onGis called a character ofGif for all

a,b∈G,

χ(a·b) =χ(a)χ(b).

In particular, ifGis a ﬁnite group with the identity elemente, thenχ(e) =  andχ(a) is a root of unity. In [, Theorem .], it is proved that a ﬁnite abelian groupGof ordernhas exactlyndistinct characters.

LetGbe the group of reduced residue classes module positive integerf. Corresponding to each characterχofG, the Dirichlet characterχDis deﬁned as follows:

χD(n) = ⎧ ⎨ ⎩

χ(n) if gcd(n,f) = ,

 if gcd(n,f) > .

Hence it is easily observed that

χ(nf +a) =χ(a)

for alln,a∈N. In this note,f is called the conductor of the character of a groupGof reduced residue classes module a positive integerf.

1.2 q-Eulerian polynomials and numbers

In [], Simsek deﬁned and studied some properties of q-Apostol type Frobenius-Euler polynomials and numbers.

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(i) Theq-Apostol type Frobenius-Euler numbers

Hn(u;a,b;λ;q) (λ,q∈C)

are deﬁned by means of the following generating function:

Fλ,q(t;u,a,b) =

 –a t

u

∞

n=

λ u

n

b[n]t= ∞

n=

Hn(u;a,b;λ;q)

tn

n!.

(ii) Theq-Apostol type Frobenius-Euler polynomials

Hn(x;u;a,b;λ;q) (λ∈C)

Fλ,q(x,t;u,a,b) =

 –a t

u

∞

n=

λ u

n

b[n+x]t= ∞

n=

Hn(x;u;a,b;λ;q)

tn

n!, ()

where λ

ub

t_{< .}

2 q-Eulerian polynomials and numbers attached to any character

In this section, we deﬁneq-Eulerian polynomials and numbers attached to any character of the ﬁnite cyclic groupG. Our new generating functions are related to nonnegative real parameters.

Deﬁnition . Leta,b∈R+₍_a₌_b_and_a_≥_),_u_∈_C_{},_λ_,_q_∈_C_(|_q_|_{< ). Let}_χ_{be a} character of a ﬁnite cyclic groupGwith the conductorf.

(i) Theq-Eulerian numbers attached to the characterχ

Hn,χ(u;a,b;λ;q)

Fλ,q,χ(t,u,a,b) =

 –a

[f]t

uf ∞

m=

λ u

m

b[m]tχ(m) = ∞

n=

Hn,χ(u;a,b;λ;q)

tn

n!. ()

(ii) Theq-Eulerian polynomials attached to the characterχ

Hn,χ(x;u;a,b;λ;q)

Fλ,q,χ(t,x,u,a,b) =

 –a

[f]t

uf ∞

m=

λ u

m

b[m+x]tχ(m) = ∞

n=

tn

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where

λ_ubt< .

It is observed that

Hn,χ(;u;a,b;λ;q) =Hn,χ(u;a,b;λ;q).

Upon settingt=  in (), we compute aq-Eulerian numberH,χ(u,a,b,λ,q) attached to

the characterχas follows:

H,χ(u,a,b,λ,q) =

u–f(uf – )

u–λχ()

.

By using the conductorfof the characterχand combining withFλ,q(t,x,u,a,b), we modify

Equation () and Equation (), respectively, as follows:

Fλ,q,χ(t,u,a,b) =

f–

i=

λχ()

u

i

Fλf_,qf

t[f],i

f,u

f_,_a_,_b

()

and

f

i=

λχ()

u

i

Fλf,qf

[f]t,i+x

f ,u

f_,_a_,_b

. ()

Therefore, we provide the following relationships betweenq-Eulerian numbers and q -Eulerian numbers attached to the characterχ.

Theorem . Let n∈N.Then we have

(i)

Hn,χ(u;a,b;λ,u) = [f]n

f–

i=

λ u

i

χ(i)Hn

i f,u

f_,_a_,_b_;_λf_,_qf

,

(ii)

Hn,χ(u;a,b;λ,u) = [f]n

f–

i=

χ(i)

λ u

i Hn

x+i f ,u

f_,_a_,_b_;_λf_,_qf

.

Proof By using (), we deduce that

∞

n=

Hn,χ(u;a,b;λ,u)

tn n!=

f–

i=

χ(i)

λ u

i∞

n= [f]nHn

i f,u

f_,_a_,_b_;_λf_,_qf

tn n!,

which, by comparing the coeﬃcient t_n!n on the both sides of the above equations, yields the ﬁrst assertion of Theorem ..

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By Theorem ., we also compute aq-Eulerian number attached to the characterχas follows:

H,χ(u,a,b,λ,q) =

f–

i=

χ(i)

λ u

i H

i f,u

f_,_a_,_b_;_λf_,_qf

.

Now, we turn our attention to the following generation function deﬁned in [] since we need this generating function frequently to give some functional equations for aq-Eulerian number and polynomials attached to the characterχ.

Leta≥. The numberYn(u,a) is deﬁned by means of the following generating function:

G(t,u,a) = 

at_–_u= ∞

n=

Yn(u,a)t n

n!.

The polynomialsYn(x;u,a) are deﬁned by means of the following generating function:

G(t,x,u,a) =G(t,u,a)axt= ∞

n=

Yn(x;u,a)t n

n!. ()

Since we need this generating function frequently in this paper, we use the notation

D(k) = a [f]t_–_uf

a[f]qk_t

–uf =G

[f]qkt, 

qk,u f_,_a

–ufG[f]qkt,uf,a

= ∞

n= [f]nqkn

Yn



qk,u f_,_a

–ufYn uf,a tn

n!,

and so it follows thatDn(k) = [f]nqn(Yn(_q,uf,a) –Yn(uf,a)). By using the following well-known identity:

[n+x] = [x] +qx[n]

in (), we verify the following functional equation:

Fλ,q,χ(t,x+y,u,a,b) =b[y]tD(y)Fλ,q,χ qyt,x,u,a,b

. ()

Hence we have the following theorem.

Theorem . Let n∈N.Then we have

Hn,χ(x+y;u;a,b;λ;q) =

m

i= i

j=

m i

i

j [y]lnb

i–j

Dj(y)q(m–i)y

×H(m–i)i,χ(x,u;a,b;λ;q).

Proof By applying the Cauchy product to (), we deduce that

Fλ,q,χ(t,x+y,u,a,b) =

∞

n= n

j=

n

j [y]lnb

n–j

Dj(y)t n

n!Fλ,q,χ tq

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LetK(n) =n_j= n_j([y]lnb)n–j_D_j(_y_{). Then it follows that}

Fλ,q,χ(t,x+y,u,a,b) =

∞

m= m

i=

m i

K(i)q(m–i)yHm–i,χ(x,u;a,b;λ;q)

tm

m!

= ∞

m= m

i= i

j=

m i

i

j [y]lnb

i–j

Dj(y)q(m–i)y

×H(m–i)i,χ(x,u;a,b;λ;q)

tm

m!.

By comparing the coeﬃcient of t_m!m on both sides of the above equation, we obtain our

desired result.

Upon settingx=  in (), we get the following functional equation:

Fλ,q,χ(t,y,u,a,b) =b[y]tD(y)Fλ,q,χ qyt, ,u,a,b

=b[y]tD(y)Fλ,q,χ qyt,u,a,b

.

By substitutingx=  in Theorem ., the following theorem is easily proved.

Theorem . Let n∈N.Then we have

Hn,χ(y;u;a,b;λ;q) =

m

i= n–i

j=

m i

i

j [y]lnb

i–j

Dj(y)q(n–i)yH(m–i)i,χ(u;a,b;λ;q).

So that we obtain a q-diﬀerence equation forq-Eulerian polynomials attached to the characterχ, we study the following equations:

 –a

[f]t

uf

+λχ()

u Fλ,q,χ(t,x+ ,u,a,b)

=

 –a [f]t

uf

+λχ()b t

u D()Fλ,q,χ(qt,x,u,a,b),

which, in light of the Cauchy product of the three seriesbt,Fλ,q,χ(qt,x,u,a,b) and

D() =G

q[f]t,

q,u

f_,_a

–ufGq[f]t,uf,a,

yield the following theorem.

Theorem . Let n∈N.Then we have

H,χ(x;u;a,b;λ;q) =  –

([f]lna)n

uf +

λχ()

u CH,χ(x;u;a,b;λ;q)

and

Hn,χ(x;u;a,b;λ;q) = –

([f]lna)n

uf +

λχ()

u

n

i=

n i

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where

Cn= n

i=

n i

(lnb)n–i[f]iqi

Yi



q,u

f_,_a

–ufYi(,u,a)

.

Now, we turn our attention to studying the derivative of the polynomials

∂

∂tFλ,q,χ(x,t,u,a,b) =

–[f]a[f]t_ln_a

uf

∞

n=

λ u

n

χ(n)b[n+x]t

+

 –a [f]t

uf ∞

n=

λ u

n

χ(n)[n+x]b[n+x]tlnb. ()

By

∞

i=

yn[n] =  ( –y)( –q)–

y

( –qy)( –q)

and substitutingt=  in (), we get that

H,χ(x;u;a,b;λ;q) =

–[f]lna uf_–_uf–_λχ_()+

uf_{– }

uf

u u–λχ()

[x]lnb

+

uf _{– }

uf_{( –}_q₎

qx

u u–λχ()+

u u–qλχ()

.

Hence, using the induction method, we arrive at the following result.

Theorem . Let n∈N.Then we have

tn

n!=

∂n

∂tnFλ,q,χ(x,t,u,a,b)

t= .

Now we give a generalization of the Raabe formula by the following theorem.

Theorem . Let c,n∈N.Then we have

Hn,χ(x;u;a,b;λ;q) =

c–

n= n

j=

n j

λχ()

u Rj(c)[c]

n–j

×Hn–j,χc

x+l c ;u

c_;_a_,_b_;_λc_;_qc

,

where

Rj(c) = [cfc]j

Yn

[f] [cfc];u

f_,_a

–ufYn uf,a

and

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Proof We start the proof with deﬁning the character

χc(x) =χ(cx)

with the conductorfc=_gcd(ff _,c). On the other hand, we derive that

uf_–_a[f]t

uf

= u f_–_a[f]t

uf_–_a[fc]qc[c]t

uf _–_a[fc]qc[c]t

uf

=R(c)u

f _–_a[fc]qc[c]t

uf ,

where

R(c) = a [f]t_–_uf

a[cfc]t_–_uf.

Then by using (), it follows that

R(c) =G

[cfc]t, [f] [cfc],u

f_,_a

–ufG [cfc]t, ,uf,a

= ∞

n= [cfc]n

Yn

[f] [cfc];u

f_,_a

–ufYn uf,a tn n! = ∞ n= Rn(c)t

n

n!.

Now, we are ready to prove our result.

∞

n=

tn

n!

=

 –a [f]t uf ∞ m= c– l= λ u l+mc

b[l+mc+x]tχ(l+mc)

=

 –a [f]t uf c– l= λχ() u

l∞

m=

λc uc

b[m+l+cx]qc[c]t_χ₍_cm₎

= c– l= λχ() u

l∞

n= Rn(c)t

n

n! ∞

n=

[c]nHn,χc

x+l c ;u

c_;_a_,_b_;_λc_;_qc tn n! = ∞ n= c– l= λχ() u

ln

j=

n j

Rj(c)[c]n–jHn–j,χc

x+l c ;u

c_;_a_,_b_;_λc_;_qc

tn n!.

Hence, comparing the coeﬃcient of tn

n! on both sides yields the assertion of this

theo-rem.

3 Decomposition of the generating function

In this section, using the generating element of the ﬁnite cyclic groupGand the generating element of the subgroups ofG, we show that the generating function with a conductorf

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In this section, we use the following notations otherwise stated: (i) xN={,x, x, x, x, . . .},

(ii) pidenotes the integer such thatgcd(pi,pj) = for alli,j∈ {, . . . ,n}, (iii) S=n_i=(piN),

(iv) S={p, . . . ,pn},

(v) Si={lcd(a,b) :a,b∈Si–}. We start to recall the fact that

xN∩yN=lcd(x,y)N

for positive integersxandy. In particular, we get that

xkN∩ytN=xyN

wheneverxandyare distinct prime numbers.

Theorem . With the above notations,we get

i∈S

F(i) = n–

j=

(–)j t∈Sj

i∈tN F(i).

Proof We start to recall the fact that for setsA,B,

i∈A∪B

F(i) = i∈A

F(i) + i∈B

F(i) – i∈A∩B

F(i).

Then by using De-Morgan’s law of sets, we get the following equality:

i∈(pN)∪(ni=(piN))

F(i) = i∈(pN)

F(i) + i∈(n

i=(piN))

F(i) – i∈(pN)∩(ni=(piN))

F(i)

=

i∈(pN)

F(i) + i∈(n

i=(piN))

F(i) – i∈(n

i=lcd(p,pi)N)

F(i).

Now we use induction onn.

Ifn= , thenS={p,p}andS={lcd(p,p)}. We have

i∈(

i=(piN))

F(i) = i∈(pN)

F(i) + i∈(pN)

F(i) – i∈lcd(p,p)N

F(i)

= 

j=

(–)j t∈Sj

i∈tN F(i).

We construct the following sets:

S= n

i= (piN),

S= n

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and

S= (pN)∪S.

By using De-Morgan’s law of sets, we verify the following equalities:

S= (pN)∩S= n

i=

lcd(p,pi)N

,

S=S∪ {p},

S=

lcd(a,b) :a,b∈S

=S_∪S_,

S=

lcd(a,b) :a,b∈S

=S_∪S_,

Si=

lcd(a,b) :a,b∈Si–

=S_i∪S_i–

and for alli≥, it is clear thatS_i∩S_i–= .

Now assume that it is true for the set withn–  elements. Hence, the hypothesis for the setsSandSis true, and we get that

i∈S

F(i) = n–

j=

(–)j t∈Sj

i∈tN F(i)

and

i∈S

F(i) = n–

j=

(–)j t∈Sj

i∈tN F(i).

Hence it follows

i∈S

F(i) = i∈pN

F(i) + i∈S

F(i) – i∈S

F(i)

=

i∈pN F(i) +

n–

j=

(–)j t∈S

j

i∈tN F(i) –

n–

j=

(–)j t∈S

j

i∈tN F(i)

=

t∈S

i∈tN

F(i) – t∈S

∪S

i∈tN F(i) +

n–

j=

(–)j t∈S

j∪Sj–

i∈tN F(i)

= n–

j=

(–)j t∈Sj

i∈tN F(i).

Example . Letp,p,pbe distinct prime integers. Then we have

i∈(

i=(piN))

F(i) = i∈(pN)

F(i) + i∈(pN)

F(i) + i∈(pN)

F(i) – i∈(ppN)

F(i)

–

i∈(ppN)

F(i) – i∈(ppN)

F(i) + i∈(pppN)

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Lemma . Let pibe a prime number for all i and

C=l∈N:gcd(f,l) = 

.

ThenN=C∪( n

i=(piN))and C∩( n

i=(piN))is an empty set.

Proof By using the fact (l,f) =  if and only if (l,n_i=pi) =  if and only if (l,pi) =  for all

i∈ {, . . . ,n}forn∈N, we get that

C=l∈N:gcd(f,l) = 

=l∈N: for alli∈ {, . . . ,n},gcd(pi,l) = 

.

Letx∈/n_i=(piN). Then we get thatgcd(x,pi) =  for alliand sox∈C. Ifx∈/C, then there isi∈ {, . . . ,n}such thatgcd(x,pi) ispi. Therefore

N=C∪ _n

i= (piN)

.

Letx∈C∩(n_i=(piN)). Then there isi∈ {, . . . ,n}such thatx∈piN. This means that

gcd(x,pi) =piand sox∈/C, a contradiction. Thus the proof is completed.

By using the Lemma ., we have one of the main results in this section.

Theorem . With the above notations,we get that

i∈N

F(i) = i∈C

F(i) + i∈(n

j=(pjN)) F(i).

As stated before, to decompose the generating function ofq-Eulerian polynomials at-tached to the characterχ, now we need to compute the following relation:

n∈(dZ)

λ u

n

b[n]tχ(n),

where

fd=

f d∈N.

Also, we deﬁne the characterχgwith the conductorhsuch as

χg(i) :=χ(gi),

whereg,h∈Nandgcd(g,h) = . Then

n∈(dZ)

λ u

n

b[n]tχ(n) = n∈N

λ u

dn

b[dn]tχ(dn)

=

n∈N

fd–

i=

λ u

d(fdn+i)

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= n∈N

fd–

i=

λd

ud fdn+i

bt[d][fdn+i]qd_χ_d(_i₎

=

i∈N

λd

ud i

bt[d][i]qd_χ_d(_i_).

Hence, we combine the above equation with the generating function

 –a

[f]t

uf

n∈(dZ)

λ u

n

b[n]tχ(n) =

 –a [dfd]t

udf i∈N

λ u

i

b[i]qdt_χ_d(_i₎

=

 –a [d][fd]qdt

(ud₎fd i∈N

λ u

i

b[i]_dqdtχd(i)

=Fλd_,qd_,_χ d t,u

d_,_a[d]_,_b[d]_.

Now we are ready to state the main result without the proof in this section.

Theorem . Let f =n_i=ptt_i,where piis a prime integer.Then

Fλ,q,χ(t,u,a,b) =

n–

j=

(–)j t∈Sj

i∈tZ

Fλi_,qi_,_χ i t,u

i_,_a[i]_,_b[i]₊_F

λ,q,χD(t,u,a,b),

whereχDis a Dirichlet character corresponding to the characterχ.

Example . Letf =pt

p t

p t

, wherepiis a prime integer. Then

Fλ,q,χ(t,u,a,b) =Fλ,q,χD(t,u,a,b) + 

i=

Fλpi,qpi,χ_pi t,upi,a[pi],b

[_pi] [_fpi]

qpi

+Fλppp_,qppp_,χp_p_p_ t,u

ppp_,_a[ppp]_,_b

[p_p_p_]

[fppp ]qppp

– 

i=

Fλppi,qppi,χppi t,u

ppi_,_a[ppi]_,_b [p__pi] [fppi]_qp__pi

–Fλpp,qpp,χpp t,u

pp_,_a[pp]_,_b [pp] [fpp ]qpp_.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors completed the paper together. Both authors read and approved the ﬁnal manuscript.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

All authors are partially supported by Research Project Oﬃces Akdeniz Universities. The author would like to thank to all referees for their valuable comments and also for suggesting references [7–14].

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doi:10.1186/1687-1812-2013-72