# An Extended Generalizedq-Extensions for the Apostol Type Polynomials

Authors: Letelier Castilla; William Ramírez; Alejandro Urieles
Journal: Abstract and Applied Analysis
 (2018)

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## Abstract

Through a modification on the parameters associated with generating function of theq-extensions for the Apostol type polynomials of order α and level m, we obtain some new results related to a unified presentation of the q-analog of the generalized Apostol type polynomials of order α and level m. In addition, we introduce some algebraic and differential properties for the q-analog of the generalized Apostol type polynomials of order α and level m and the relation of these with the q-Stirling numbers of the second kind, the generalized q-Bernoulli polynomials of level m, the generalized q-Apostol type Bernoulli polynomials, the generalized q-Apostol type Euler polynomials, the generalized q-Apostol type Genocchi polynomials of order α and level m, and the q-Bernstein polynomials.


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## Body

## 1. Introduction

With the development ofq-calculus in the mid-19th century, many authors made generalizations to special functions and polynomial families based on the q-analogs (cf. [1–7]). During this process, properties and relations have been demonstrated and contributed to solving different kinds of problems in other subjects (see, [8–10]).In 2003, Natalini P. and Bernardini A. [11] introduced a class the polynomials Bn[m-1](x), considering a class of Appell polynomials defined by using a generating function linked to the Mittag-Leffler function (see, [12, p. 204, Eq. (2)])(1)E1,m+1z≔zmez-∑h=0m-1zh/h!,m∈N.Kurt B. [13] did the generalization of the Bernoulli Bn[m-1,α](x), Euler En[m-1,α](x), and Genocchi Gn[m-1,α](x) polynomials of order α and level m. Tremblay R. et al. [14, 15] defined the generalized Apostol-Bernoulli polynomials Bn[m-1,α](x;λ) and their properties. In [12] which has studied the unification of Bernoulli, Euler, and Genocchi polynomials they considered the following Mittag-Leffler type function (see, [12, p. 209, Eq. (12)]):(2)E1,m+1c,a;λ;u;vz≔2uzvmλcz+∑l=0m-1zlnal/l!,m∈N,a,c∈R+,λ,μ,ν∈C,to define a extension of the generalized Apostol type polynomials in x, parameters c,a,μ,ν, order α, and level m through the following generating function:(3)E1,m+1c,a;λ;u;vzαcxz≔∑n=0∞Qnm-1,αx;c,a;λ,μ;νznn!,where m∈N, a,c∈R+, and α,λ,μ,ν∈C. The numbers were given by (4)Qnm-1,αc,a;λ,μ;ν≔Qnm-1,α0;c,a;λ,μ;ν.Recently a new class of polynomials has been introduced in [16] and it provides a unification of three families of polynomials through the following generating function (see, [16, p. 923, Eq. (3)]): (5)21-kzkβbeqz-abαeqxzEqyz=∑n=0∞Pn,β,qαx,y;k,a,bznnq!,where k∈N0, a,b∈R∖0, and α,β∈C. The author named it the unified q-Apostol-Bernoulli, Euler, and Genocchi polynomials of order α and proved some properties for these unification.On the other hand, in the definition of theq-Mittag-Leffler function when α=1, β=m+1, and γ=1 lead us to (see, [17, p. 614, Eq (1.5)])(6)E1,m+1z;q≔zmeqz-∑h=0m-1zh/hq!,m∈N,which corresponds to the q-analog of the generating function defined in (1) and with this, new research emerged about other polynomial families based on the q-analogs.In [18], the authors introduced the generalized q-Apostol-Bernoulli polynomials, the generalized q-Apostol Euler polynomials, and generalized q-Apostol Genocchi polynomials in variable x,y, order α, and level m through the following generating functions, defined in a suitable neighborhood of z=0 (see, [18, p. 2 Eq (8), (9), (10)])(7)zmλeqz-Tm-1,qzαeqxzEqyz=∑n=0∞Bn,qm-1,αx,y;λznnq!,(8)2mλeqz+Tm-1,qzαeqxzEqyz=∑n=0∞En,qm-1,αx,y;λznnq!,(9)2mzmλeqz+Tm-1,qzαeqxzEqyz=∑n=0∞Gn,qm-1,αx,y;λznnq!,where q,α∈C, m∈N, 0<q<1, and Tm-1,q(z)=eqz-∑h=0m-1zh/hq!.Based on the previous result, we focus our attention on a new unification of theq-analog of the generalized Apostol type polynomials of order α and level m, defined in [18], by doing some modifications to the generating function linked to the q-Mittag-Leffler function (6) through new parameters following the same scheme or procedure applied by [12].The paper is organized as follows. Section2 contains some notations, definitions, and properties of the q-analogs and some results about q-analogs of the Apostol type polynomials. In Section 3, we introduce the unification q-analog of the generalized Apostol type polynomials in x,y, parameters λ,μ,ν∈C, order α∈C, and level m∈N and their algebraic and differential properties. Finally in the Section 4, we show relations between the q-analog of the generalized Apostol type polynomials and the q-Stirling numbers of the second kind, the generalized q-Bernoulli polynomials of level m, the generalized q-Apostol type Bernoulli polynomials, the generalized q-Apostol type Euler polynomials, the generalized q-Apostol type Genocchi polynomials of order α and level m, and the q-Bernstein polynomials.


## 2. Background and Previous Results

Throughout this paper, we use the following standard notions:N=1,2,…, N0=0,1,2,…, Z denotes the set of integers, R denotes the set of real numbers, and C denotes the set of complex numbers. The q-numbers and q-factorial numbers are defined, respectively, by (10)zq=1-qz1-q=qz-1q-1,z∈C,q∈C∖1,qz≠1,nq!=∏k=1nkq=1q2q3q…nq,0q!=1,n∈N.El q-shifted factorial is defined as (11)a;qn=∏j=0n-11-qja,n∈N.The q-binomial coefficient is defined by (12)nkq=nq!n-kq!kq!,n,k∈N0;0≤k≤n. For more information about the q-standard definitions and properties see [8, 9, 19].Furthermore, theq-binomial coefficient satisfies the following identity (see [10, p. 483, Eq. (41)]):(13)nkqkjq=njqn-jk-jq0≤j≤k.The q-analog of the function x+yn is defined by (14)x+yqn=∑k=0nnkqq1/2kk-1xn-kyk,n∈N0,1-aqn=a;qn=∑k=0nnkqq1/2kk-1-1kak=∏j=0n-11-qja.The q-derivative of a function f(z) is defined by (15)Dqfz=dqfzdqz=fqz-fzq-1z,0<q<1,0≠z∈C. The q-analog of the exponential function is defined in two ways (16)eqz=∑n=0∞znnq!=∏k=0∞11-1-qqkz,0<q<1,z∈C,z<11-q,Eqz=∑n=0∞q1/2nn-1znnq!=∏n=0∞1+1-qqkz,0<q<1,z∈C;we can see that(17)eqz·Eq-z=1,(18)eqz+w=eqz·Eqw.Therefore, (19)Dqeqz=eqz,DqEqz=Eqqz.For anyt>0 (see, e.g., [19, p. 76, Eq (21.6)]) (20)Γqt=∫0∞xt-1Eq-qxdqxis called the q-gamma function.The Jackson’sq-gamma function is defined by (see [10, p. 490, Eq. (2)])(21)Γqx=q;q∞qx;q∞1-q1-x,0<q<1.In (21) we have (22)Γqn+1=q;q∞qn+1;q∞1-q-n=q;qn1-q-n=nq!,n∈N.Forα,β,γ∈C, Re(α)>0, Re(β)>0, and Re(γ)>0 y q<1 the function Eα,βγ(z;q) is defined as (see, e.g., [17, p. 614, Eq. (1.5)]) (23)Eα,βγz;q=∑n=0∞qγ;qnq;qnznΓqαn+β.Notice that, when γ=1 the previous equation is expressed as (24)Eα,βz;q=∑n=0∞znΓqαn+β.Settingα=1 y β=m+1, we can deduce(25)∑n=0∞znΓqn+m+1=∑n=0∞znn+mq!=1zm∑h=m∞zhhq!=eqz-∑h=0m-1zh/hq!zm.The q-Stirling numbers of the second kind S(n,k)q are defined through the following expansion (see, e.g., [20, p. 173, Equ (5.18)]):(26)xn=∑k=0nSn,kqxk,q,where (27)xk,q=∏n=0k-1x-nq.LetC[0,1] denote the set of continuous function on [0,1]. For any f∈C[0,1], the q- Bn(f;x) is called q-Bernstein operator of order n for f and is defined as (see, e.g., [21, p. 2, Eq (1.1)]) (28)Bnf;x=∑r=0nfrnrqxr∏s=0n-r-11-qsx=∑r=0nfrBrnx, where fr=f[r]q/[n]q. For n,r∈N0, the q-Bernstein polynomials of degree n or q-Bernstein basis are defined by (29)Brnx=nrqxr∏s=0n-r-11-qsx.We know that ∑k=0n-jBkn-j(x)=1, then (30)xj=∑k=0n-jn-jkqxj+k∏t=0n-j-k-11-qtx, and using the identity (13), we have (see [21, p. 6, Eq (2.3)])(31)xj=∑k=jnkjqnjqBknx.Mahmudov N.I. [22] made a relation between the q-Bernstein basis with the q-Stirling numbers of the second kind and the q-Bernoulli polynomials of order α, α=k as follows:(32)Bknx;q=xk∑j=0nnjqSj,k;qBn-jk1,-x;q.Proposition 1.
For a fixedm∈N, n,k∈N0, and 0<q<1, let Bnm-1,α(x,y;q)n=0∞ be the sequence of generalized q-Bernoulli polynomials in x,y of level m. Then the following identities are satisfied:
(1) [23, Lemma 10, Eq. (1)](33)xn=∑k=0nnkqkq!k+mq!Bn-km-1,1x,0;q.
(2) [23, Lemma 10, Eq. (2)](34)yn=1q1/2nn-1∑k=0nnkqkq!k+mq!Bn-km-1,10,y;q.
Now, settingk=n-k, we have(35)yn=nq!q1/2nn-1n+mq!∑k=0nn+mkqBkm-1,10,y;q.
(3)(36)xn=∑k=0nnq!Bkm-1,1x,0;qkq!Γqn-k+m+1.Proof (see (36)).
Settingα=λ=1, y=0 in (7) and using (25), we have (37)eqxz∑n=0∞zn/Γqn+m+1=∑n=0∞Bnm-1,1x,0;qznnq!eqxz=∑n=0∞Bnm-1,1x,0;qznnq!∑n=0∞znΓqn+m+1=∑n=0∞∑k=0nnq!Bkm-1,1x,0;qkq!Γqn-k+m+1·znnq!∑n=0∞xnznnq!=∑n=0∞∑k=0nnq!Bkm-1,1x,0;qkq!Γqn-k+m+1·znnq!.Comparing the coefficients of zn/nq! we obtain(38)xn=∑k=0nnq!Bkm-1,1x,0;qkq!Γqn-k+m+1.Notice that in (36), as Γq(n-k+m+1)=[n-k+m]q! we can get (33).Based on the results of (2), (6), and (7)–(9) and following the methodology given in [12], we consider the following q-Mittag-Leffler type function:(39)E1,m+1λ;μ;νz;q≔2μzνmλeqz+∑h=0m-1zh/hq!,where m∈N, λ,μ,ν∈C.


## 3. The PolynomialsTnm-1,α(x,y;λ;μ;ν;q) and Their Properties

Definition 2.
Letm∈N, n∈N0, α,λ,μ,ν,q∈C, and 0<q<1; the q-analog of the generalized Apostol type polynomials in x,y, with parameters λ,μ,ν, order α, and level m is defined by means of the following generating function, in a suitable neighborhood of z=0,(40)2μzνmλeqz+∑l=0m-1zl/lq!αeqxzEqyz=∑n=0∞Tnm-1,αx,y;λ;μ;ν;qznnq!,where z<logq⁡(-λ) when λ∈C∖-1,1 and 1α≔1. The numbers are given by(41)2μzνmλeqz+∑l=0m-1zl/lq!α=∑n=0∞Tnm-1,αλ;μ;ν;qznnq!.Notice that(42)Tnm-1,α0,0;λ;μ;ν;q≔Tnm-1,αλ;μ;ν;q.Therefore, (43)Tnm-1,αx,0;λ;μ;ν;q≔Tnm-1,αx,λ;μ;ν;q,Tnm-1,α0,y;λ;μ;ν;q≔Tnm-1,αy,λ;μ;ν;q.We will use this notation Tn[m-1,α](λ;μ;ν;q) instead of Tn,q[m-1,α](λ;μ;ν) through the article. By comparing Definition 2 with (7)–(9), we have (44)-1αTnm-1,αx,y;-λ;0;1;q=Bn,qm-1,αx,y;λ,Tnm-1,αx,y;λ;1;y;q=En,qm-1,αx,y;λ,Tnm-1,αx,y;λ;1;1;q=Gn,qm-1,αx,y;λ.Therefore, (45)limq→1-⁡Tnm-1,αx,y;λ;μ;ν;q=Tnm-1,αx+y;λ,μ;ν,limq→1-⁡Tnm-1,α0,0;λ;μ;ν;q=Tnm-1,αλ,μ;ν.Clearly for m=1, we have (46)Tn0,αx,y;λ;μ;ν;q=Tnαx,y;λ;μ;ν;q.For α=1, we have (47)Tnm-1,1x,y;λ;μ;ν;q=Tnm-1x,y;λ;μ;ν;q.For m=α=1, we have (48)Tn0,1x,y;λ;μ;ν;q=Tnx,y;λ;μ;ν;q.Example 3.
Whenα=1, μ=0, and ν=1 we can take λ=-1. (49)-T0m-1,1x,0;-1;0;1;q=B0m-1x;q=mq!,-T1m-1,1x,0;-1;0;1;q=B1m-1x;q=mq!x-1m+1q,-T2m-1,1x,0;-1;0;1;q=B2m-1x;q=mq!x2-2qxm+1q+2qqm+1m+2qm+1q2.And the numbers are as follows:(50)-T0m-1,10,0;-1;0;1;q=B0m-1q=mq!,-T1m-1,10,0;-1;0;1;q=B1m-1q=-mq!m+1q,-T2m-1,10,0;-1;0;1;q=B2m-1q=2qmq!qm+1m+2qm+1q2.Example 4.
For anyλ∈C∖-1,1, m=1, α=1, μ=1, and ν=2(51)Tj0,1x,0;λ;1;2;q=0,j=0,1,T20,1x,0;λ;1;2;q=22qλ+1,T30,1x,0;λ;1;2;q=23q!xλ+1-23q!λλ2+2λ+1.Example 5.
For anyλ∈C∖-1,1, m=2, α=1, μ=3, and ν=1(52)Tj1,1x,0;λ;3;1;q=0,j=0,1,T21,1x,0;λ;3;1;q=642qλ+1,T31,1x,0;λ;3;1;q=643q!xλ+1-643q!λ+1.The following proposition summarizes some properties of the polynomialsTn[m-1,α](x,y;λ;μ;ν;q) which are a consequence of (40). Therefore, we will only show the details of its proofs (7) and (8).Proposition 6.
For a fixedm∈N, 0<q<1, let Tnm-1,α(x,y;λ;μ;ν;q)n=0∞ be the sequence of the q-analog of the generalized Apostol type polynomials in x,y, parameters α,λ,μ,ν∈C, order α, and level m. Then the following statements hold:
(1) Special values: for everyn∈N0(53)Tnm-1,0x,0;λ;μ;ν;q=xn,(54)Tnm-1,00,y;λ;μ;ν;q=q1/2nn-1yn.
(2) Summation formulas: for everyn∈N0(55)Tnm-1,αx,0;λ;μ;ν;q=∑k=0nnkqTn-km-1,αλ;μ;ν;qxk,(56)Tnm-1,α0,y;λ;μ;ν;q=∑k=0nnkqq1/2kk-1Tn-km-1,αλ;μ;ν;qyk,(57)Tnm-1,αx,0;λ;μ;ν;q=∑k=0nnkqTn-km-1,α-1λ;μ;ν;qTkm-1,1x,0;λ;μ;ν;q,(58)Tnm-1,αx,0;λ;μ;ν;q=∑k=0nnkqTn-km-1,αλ;μ;ν;qTkm-1,0x,0;λ;μ;ν;q.
(3) Differential relations: form∈N, fixed α,λ and n,j∈N0 with 0≤j≤n, we have(59)DqTn+1m-1,αx,0;λ;μ;ν;q=n+1qTnm-1,αx,0;λ;μ;ν;q,(60)DqjTnm-1,αx,0;λ;μ;ν;q=nq!n-jq!Tn-jm-1,αx,0;λ;μ;ν;q.
(4) Integral formulas: form∈N, fixed α,λ, we have(61)∫x0x1Tnm-1,αx,0;λ;μ;ν;qdqx=Tn+1m-1,αx1,0;λ;μ;ν;q-Tn+1m-1,αx0,0;λ;μ;ν;qn+1q,(62)∫x0x1Tnm-1,αx,0;λ;μ;ν;qdqx=∑k=0nnkqTkm-1,αλ;μ;ν;qx1n+1-k-x0n+1-kn+1-kq,(63)Tnm-1,αx,0;λ;μ;ν;q=nq∫0xTn-1m-1,αx,0;λ;μ;ν;qdqx+Tnm-1,αλ;μ;ν;q.
(5) Addition theorems:(64)Tnm-1,αx,y;λ;μ;ν;q=∑k=0nnkqq1/2kk-1Tn-km-1,αx,0;λ;μ;ν;qyk,(65)Tnm-1,αx,y;λ;μ;ν;q=∑k=0nnkqq1/2kk-1Tn-km-1,α0,y;λ;μ;ν;qxk,(66)Tnm-1,αx,y;λ;μ;ν;q=∑k=0nnkqTn-km-1,αλ;μ;ν;qx+yqk,(67)Tnm-1,αx,y;λ;μ;ν;q=∑k=0nnkqTn-km-1,α0,y;λ;μ;ν;qxk,(68)Tnm-1,α+βx,y;λ;μ;ν;q=∑k=0nnkqTn-km-1,αx,0;λ;μ;ν;qTkm-1,β0,y;λ;μ;ν;q.
Clearly, settingx=0 in (64), we have (69)Tnm-1,α0,y;λ;μ;ν;q=∑k=0nnkqq1/2kk-1Tn-km-1,αλ;μ;ν;qyk.
Settingy=0 in (67), we obtain (55) (70)Tnm-1,αx,0;λ;μ;ν;q=∑k=0nnkqTn-km-1,αλ;μ;ν;qxk.
Settingy=1 and x=1 in (64) and (67), respectively, we have (71)Tnm-1,αx,1;λ;μ;ν;q=∑k=0nnkqq1/2kk-1Tn-km-1,αx,0;λ;μ;ν;q,Tnm-1,α1,y;λ;μ;ν;q=∑k=0nnkqTn-km-1,α0,y;λ;μ;ν;q.
(6) Ifa∈N, we have(72)Tnm-1,αx,ax-x;λ;μ;ν;q=∑k=0nnkqq1/2kk-1Tn-km-1,αx,0;λ;μ;ν;qa-1kxk.
(7) Theq-analog of the generalized Apostol type polynomials satisfies the following relations:(73)λTnm-1,α1,y;λ;μ;ν;q+Tnm-1,α0,y;λ;μ;ν;q=2∑k=0nnkqTkm-1,α0,y;λ;μ;ν;q×Tn-k0,-1λ;1;0;q,(74)λTnm-1,α1,y;λ;μ;ν;q+∑k=0minn,m-1nkqTn-km-1,α0,y;λ;μ;ν;q=2μm∑k=0nnkqTkm-1,α0,y;λ;μ;ν;qTn-km-1,-1λ;μ;0;q,(75)λTnm-1,α1,y;λ;μ;ν;q+∑k=0min⁡n,m-1nkqTn-km-1,α0,y;λ;μ;ν;q=2μmnq!n-νmq!Tn-νmm-1,α-10,y;λ;μ;ν;q,n≥νm,(76)λTnm-1,αx,0;λ;μ;ν;q+∑k=0min⁡n,m-1nkqTn-km-1,αx,-1;λ;μ;ν;q=2μmnq!n-νmq!Tn-νmm-1,α-1x,-1;λ;μ;ν;q,n≥νm.Proof (see (73)). Considering the expression λTn[m-1,α](1,y;λ;μ;ν;q)+Tn[m-1,α](0,y;λ;μ;ν;q) and using (40), we have(77)I≔∑n=0∞λTnm-1,α1,y;λ;μ;ν;q+Tnm-1,α0,y;λ;μ;ν;qznnq!=∑n=0∞λTnm-1,α1,y;λ;μ;ν;qznnq!+∑n=0∞Tnm-1,α0,y;λ;μ;ν;qznnq!=λ2μzνmλeqz+∑l=0m-1zl/lq!αeqzEqyz+2μzνmλeqz+∑l=0m-1zl/lq!αEqyz.Now, factoring the previous expression, we get (78)I=22μzνmλeqz+∑l=0m-1zl/lq!αEqyzλeqz+12=2∑n=0∞Tnm-1,α0,y;λ;μ;ν;qznnq!∑n=0∞Tn0,-1λ;1;0;qznnq!=∑n=0∞2∑k=0nnkqTkm-1,α0,y;λ;μ;ν;qTn-k0,-1λ;1;0;qznnq!.Comparing the coefficients of zn/[n]q! in both sides gives the result.This completes the proof.Proof (see (74)). By using the relation (see [23, p.5, Lemma 6])(79)∑l=0m-1zllq!∑n=0∞Tnm-1,α0,y;λ;μ;ν;qznnq!=∑n∞∑k=0min⁡n,m-1nkqTn-km-1,α0,y;λ;μ;ν;qznnq!and (40), we have (80)∑n=0∞λTnm-1,α1,y;λ;μ;ν;q+∑k=0minn,m-1nkqTn-km-1,α0,y;λ;μ;ν;qznnq!=∑n=0∞λTnm-1,α1,y;λ;μ;ν;qznnq!+∑l=0m-1zllq!∑n=0∞Tnm-1,α0,y;λ;μ;ν;qznnq!=λ2μzνmλeqz+∑l=0m-1zl/lq!αeqzEqyz+∑l=0m-1zllq!2μzνmλeqz+∑l=0m-1zl/lq!αEqyz Now, factoring the above expression, we get (81)∑n=0∞λTnm-1,α1,y;λ;μ;ν;q+∑k=0minn,m-1nkqTn-km-1,α0,y;λ;μ;ν;qznnq!=2μm2μzνmλeqz+∑l=0m-1zl/lq!αEqyzλeqz+∑l=0m-1zl/lq!2μm=2μm∑n=0∞Tnm-1,α0,y;λ;μ;ν;qznnq!∑n=0∞Tnm-1,-1λ;μ;0;qznnq!=2μm∑n=0∞∑k=0nnkqTkm-1,α0,y;λ;μ;ν;qTn-km-1,-1λ;μ;0;qznnq!.Comparing the coefficients of zn/[n]q! in both sides gives the result.This completes the proof.Proof (see (75)). Let (82)I≔∑n=0∞λTnm-1,α1,y;λ;μ;ν;q+∑k=0minn,m-1nkqTn-km-1,α0,y;λ;μ;ν;qznnq!.Using (40) and (79), we have (83)I=∑n=0∞λTnm-1,α1,y;λ;μ;ν;qznnq!+∑n=0∞∑k=0minn,m-1nkqTn-km-1,α0,y;λ;μ;ν;qznnq!=∑n=0∞λTnm-1,α1,y;λ;μ;ν;qznnq!+∑l=0m-1zllq!∑n=0∞Tnm-1,α0,y;λ;μ;ν;qznnq!=λ2μzνmλeqz+∑l=0m-1zl/lq!αeqzEqyz+∑l=0m-1zllq!2μzνmλeqz+∑l=0m-1zl/lq!αEqyz=2μzνm2μzνmλeqz+∑l=0m-1zl/lq!αEqyzλeqz+∑l=0m-1zl/lq!2μzνm=2μmzνm2μzνmλeqz+∑l=0m-1zl/lq!α-1Eqyz.Therefore, we get (84)I=2μm∑n=0∞Tnm-1,α-10,y;λ;μ;ν;qzn+νmnq!=2μm∑n=0∞n+νmq!nq!Tnm-1,α-10,y;λ;μ;ν;qzn+νmn+νmq!=2μm∑n-νm=0∞nq!n-νmq!Tn-νmm-1,α-10,y;λ;μ;ν;qznnq!.Notice that (85)T0m-1,α-10,y;λ;μ;ν;q=0,…,Tn-1-νmm-1,α-10,y;λ;μ;ν;q=0.Comparing the coefficients of zn/[n]q! in both sides gives the result.This completes the proof.Proof (see (76)). Let (86)I≔∑n=0∞λTnm-1,αx,0;λ;μ;ν;q+∑k=0minn,m-1nkqTn-km-1,αx,-1;λ;μ;ν;qznnq!. Using (40) and (79), we have (87)I=∑n=0∞λTnm-1,αx,0;λ;μ;ν;qznnq!+∑n=0∞∑k=0minn,m-1nkqTn-km-1,αx,-1;λ;μ;ν;qznnq!=∑n=0∞λTnm-1,αx,0;λ;μ;ν;qznnq!+∑l=0m-1zllq!∑n=0∞Tnm-1,αx,-1;λ;μ;ν;qznnq!=λ2μzνmλeqz+∑l=0m-1zl/lq!αeqxz+∑l=0m-1zllq!2μzνmλeqz+∑l=0m-1zl/lq!αeqxzEq-z.then, factoring the above equation and using (17), we have (88)I=2μmzνm2μzνmλeqz+∑l=0m-1zl/lq!αeqxzλeqz+∑l=0m-1zl/lq!2μzνm·1eqz=2μmzνm2μzνmλeqz+∑l=0m-1zl/lq!α-1eqxzEq-z. Therefore, we get (89)I=2μm∑n=0∞Tnm-1,α-1x,-1;λ;μ;ν;qzn+νmnq!=2μm∑n=0∞n+νmq!nq!Tnm-1,α-1x,-1;λ;μ;ν;qzn+νmn+νmq!=2μm∑n-νm=0∞nq!n-νmq!Tn-νmm-1,α-1x,-1;λ;μ;ν;qznnq!. Notice that (90)T0m-1,α-1x,-1;λ;μ;ν;q=0,…,Tn-1-νmm-1,α-1x,-1;λ;μ;ν;q=0.Comparing the coefficients of zn/[n]q! in both sides gives the desired result.This completes the proof.Remarks 7. Setting μ=1, ν=0, λ=1 in (75), we obtain(91)Tnm-1,α1,y;1;1;0;q+∑k=0min⁡n,m-1nkqTn-km-1,α0,y;1;1;0;q=2mTnm-1,α-10,y;1;1;0;q,Note that (91) is equivalent to [23, Lemma 6, Eq.2].Substitutingν=0 in (75), we obtain(92)λTnm-1,α1,y;λ;μ;0;q+∑k=0minn,m-1nkqTn-km-1,α0,y;λ;μ;0;q=2μmTnm-1,α-10,y;λ;μ;0;q.(8) For n ∈ N 0 , n ≥ ν m , we have (93) y n - v m = n - v m q ! 2 μ m n q ! q 1 / 2 n - v m n - v m - 1 × λ T n m - 1,1 1 , y ; λ ; μ ; ν ; q + ∑ k = 0 m i n n , m - 1 n k q T n - k m - 1,1 0 , y ; λ ; μ ; ν ; q .Proof (see (93)). Putting α=1 in (75) and using (54), we obtain (94)λTnm-1,11,y;λ;μ;ν;q+∑k=0minn,m-1nkqTn-km-1,10,y;λ;μ;ν;q=2μmnq!n-νmq!Tn-νmm-1,00,y;λ;μ;ν;q=2μmnq!n-νmq!q1/2n-νmn-νm-1yn-νm,then(95)yn-vm=n-vmq!2μmnq!q1/2n-vmn-vm-1×λTnm-1,11,y;λ;μ;ν;q+∑k=0minn,m-1nkqTn-km-1,10,y;λ;μ;ν;q.This completes the proof.


## 4. Some Connection Formulas for the PolynomialsTnm-1,α(x,y;λ;μ;ν;q)

In this section, we introduce some formulas of connection between theq-analog of the generalized Apostol type polynomials Tnm-1,α(x,y;λ;μ;ν;q) and the generalized q-Bernoulli polynomials of level m, the q-Stirling numbers of the second kind, the generalized q-Apostol type Bernoulli, q-Apostol type Euler, q-Apostol type Genocchi polynomials of order α and level m, and the q-Bernstein polynomials.Proposition 8.
Forn,j,k∈N0, m∈N, and 0<q<1, the q-analog of the generalized Apostol type polynomials of level m is related to the generalized q-Bernoulli polynomials of level m and the q-gamma function(96)Tnm-1,αx,y;λ;μ;ν;q=∑j=0n∑k=0jnjqjkqkq!k+mq!q1/2jj-1Tn-jm-1,α0,y;λ;μ;ν;qBj-km-1,1x,0;q,(97)Tnm-1,αx,y;λ;μ;ν;q=∑j=0n∑k=0jnjqjq!q1/2jj-1kq!Tn-jm-1,α0,y;λ;μ;ν;qBkm-1,1x,0;qΓqj-k+m+1,(98)Tnm-1,αx,y;λ;μ;ν;q=∑j=0n∑k=0jnjqjq!Tn-jm-1,αx,0;λ;μ;ν;qBkm-1,10,y;qj+m-kq!kq!.Proof.
We only prove (97). Substituting (36) into the right-hand side of (65), we have (99)Tnm-1,αx,y;λ;μ;ν;q=∑j=0nnjqq1/2jj-1Tn-jm-1,α0,y;λ;μ;ν;q∑k=0jjq!Bkm-1,1x,0;qkq!Γqj-k+m+1=∑j=0n∑k=0jnjqq1/2jj-1Tn-jm-1,α0,y;λ;μ;ν;qjq!Bkm-1,1x,0;qkq!Γqj-k+m+1=∑j=0n∑k=0jnjqjq!q1/2jj-1kq!Tn-jm-1,α0,y;λ;μ;ν;qBkm-1,1x,0;qΓqj-k+m+1.Proposition 9.
Forn,j,k∈N0, m∈N, and 0<q<1, the q-analog of the generalized Apostol type polynomials is related to the q-Stirling numbers of the second kind S(n,k;q) by means the following identities:(100)Tnm-1,αx,y;λ;μ;ν;q=∑j=0n∑k=0jnjqq1/2jj-1Tn-jm-1,α0,y;λ;μ;ν;qSj,k;qxq;k,(101)Tnm-1,αx,0;λ;μ;ν;q=∑j=0n∑k=0jnjqTn-jm-1,αλ;μ;ν;qSj,k;qxq;k.Proof.
Substituting (26) into the right-hand side of (55) and (65) gives the results.Proposition 10.
Forn,k∈N0, m∈N, and 0<q<1, the q-analog of the generalized Apostol type polynomials is related to the generalized q-Apostol-Bernoulli polynomials, the generalized q-Apostol-Euler polynomials, and the generalized q-Apostol-Genocchi polynomials by means the following identities:(102)λTnm-1,α1,y;λ;μ;ν;q-∑k=0min⁡n,m-1nkqTn-km-1,α0,y;λ;μ;ν;q=∑k=0n-mn-mkqnq!n-mq!Tkm-1,α0,y;λ;μ;ν;qBn-m-km-1,-1λ;q,n≥m,(103)λTnm-1,α1,y;λ;μ;ν;q+∑k=0minn,m-1nkqTn-km-1,α0,y;λ;μ;ν;q=2m∑k=0nnkqTkm-1,α0,y;λ;μ;ν;qzkkq!En-km-1,-1λ;q,(104)λTnm-1,α1,y;λ;μ;ν;q+∑k=0min⁡n,m-1nkqTn-km-1,α0,y;λ;μ;ν;q=2m∑k=0n-mn-mkqnq!n-mq!Tkm-1,α0,y;λ;μ;ν;qGn-m-km-1,-1λ;q,n≥m.We will only show the details of the demonstrations of (102) and (103).Proof (see (102)).
Using (40) and (79), we have (105)I≔∑n=0∞λTnm-1,α1,y;λ;μ;ν;q-∑k=0minn,m-1nkqTn-km-1,α0,y;λ;μ;ν;qznnq!=∑n=0∞λTnm-1,α1,y;λ;μ;ν;qznnq!-∑l=0m-1zllq!∑n=0∞Tnm-1,α0,y;λ;μ;ν;qznnq!=λ2μzνmλeqz+∑l=0m-1zl/lq!αeqzEqyz-∑l=0m-1zllq!2μzνmλeqz+∑l=0m-1zl/lq!αEqyz. Now, factoring the above equation and using (7), (40), we get (106)I=zm2μzνmλeqz+∑l=0m-1zl/lq!αEqyzλeqz-∑l=0m-1zl/lq!zm=∑n=0∞Tnm-1,α0,y;λ;μ;ν;qznnq!∑n=0∞Bnm-1,-1λ;qzn+mnq!=∑n=0∞∑k=0nTkm-1,α0,y;λ;μ;ν;qzkkq!Bn-km-1,-1λ;qzn-k+mn-kq!.Then, we get (107)I=∑n=m∞∑k=0n-mTkm-1,α0,y;λ;μ;ν;qzkkq!Bn-m-km-1,-1λ;qzn-kn-m-kq!=∑n=m∞∑k=0n-mn-mkqnq!n-mq!Tkm-1,α0,y;λ;μ;ν;qBn-m-km-1,-1λ;qznnq!. Comparing the coefficients of zn/[n]q! in both sides gives the result.Proof (see (103)).
Using (40) and (79), we have (108)I≔∑n=0∞λTnm-1,α1,y;λ;μ;ν;q+∑k=0minn,m-1nkqTn-km-1,α0,y;λ;μ;ν;qznnq!=∑n=0∞λTnm-1,α1,y;λ;μ;ν;qznnq!+∑l=0m-1zllq!∑n=0∞Tnm-1,α0,y;λ;μ;ν;qznnq!=λ2μzνmλeqz+∑l=0m-1zl/lq!αeqzEqyz+∑l=0m-1zllq!2μzνmλeqz+∑l=0m-1zl/lq!αEqyz. Now, factoring the previous equation and using (8), (40), we get (109)I=2m2μzνmλeqz+∑l=0m-1zl/lq!αEqyzλeqz+∑l=0m-1zl/lq!2m=2m∑n=0∞Tnm-1,α0,y;λ;μ;ν;qznnq!∑n=0∞Enm-1,-1λ;qznnq!=2m∑n=0∞∑k=0nTkm-1,α0,y;λ;μ;ν;qzkkq!En-km-1,-1λ;qzn-kn-kq!. Therefore, we get (110)I=∑n=0∞2m∑k=0nnkqTkm-1,α0,y;λ;μ;ν;qzkkq!En-km-1,-1λ;qznnq!. Comparing the coefficients of zn/[n]q! in both sides gives the result.Proposition 11.
Forn,j,k∈N0, m∈N, and 0<q<1, the q-analog of the generalized Apostol type polynomials is related to q-Bernstein basis Bkn(x;q) by means of the following identities: (111)Tnm-1,αx,0;λ;μ;ν;q=∑j=0n∑k=0n-jk+jjqTn-jm-1,αλ;μ;ν;qBk+jnx;q,(112)Tnm-1,αx,y;λ;μ;ν;q=∑j=0n∑k=0n-jk+jjqq1/2jj-1Tn-jm-1,α0,y;λ;μ;ν;qBk+jnx;q.Proof (see (111)).
Substituting (31) into (55) we have (113)Tnm-1,αx,0;λ;μ;ν;q=∑j=0nnjqTn-jm-1,αλ;μ;ν;q∑k=jnkjqnjqBknx;q=∑j=0n∑k=jnkjqTn-jm-1,αλ;μ;ν;qBknx;q=∑j=0n∑k=0n-jk+jjqTn-jm-1,αλ;μ;ν;qBk+jnx;q.Proof (see (112)).
Substituting (31) into (65) we have (114)Tnm-1,αx,y;λ;μ;ν;q=∑j=0nnjqq1/2jj-1Tn-jm-1,α0,y;λ;μ;ν;q∑k=jnkjqnjqBknx;q=∑j=0n∑k=jnkjqq1/2jj-1Tn-jm-1,α0,y;λ;μ;ν;qBknx;q=∑j=0n∑k=0n-jk+jjqq1/2jj-1Tn-jm-1,α0,y;λ;μ;ν;qBk+jnx;q.Proposition 12.
Forn,k,j∈N0, m∈N, and 0<q<1(115)Bknx;q=xk∑j=0n-kn-kjqnkqTn-k-jm-1,-1λ;μ;ν;qTjm-1,11,-x;λ;μ;ν;q.Proof (see (115)).
Theq-Bernstein basis is defined by means of following generating function: (116)xkzkkq!eqzEq-xz=∑n=k∞Bknx;qznnq!.Using the left-hand side of the previous equation and (40), we have (117)xkzkkq!eqzEq-xz=xkzkkq!eqz-∑h=0m-1zh/hq!2μzνm2μzνmeqz-∑h=0m-1zh/hq!eqzEq-xz=xkzkkq!∑n=0∞Tnm-1,-1λ;μ;ν;qznnq!∑j=0∞Tjm-1,11,-x;λ;μ;ν;qzjjq!=xkkq!∑n=0∞Tnm-1,-1λ;μ;ν;qzn+knq!∑j=0∞Tjm-1,11,-x;λ;μ;ν;qzjjq!.Therefore, we obtain (118)xkzkkq!eqzEq-xz=xkkq!∑n=k∞∑j=0n-kTn-k-jm-1,-1λ;μ;ν;qzn-jn-k-jq!Tjm-1,11,-x;λ;μ;ν;qzjjq!=∑n=k∞xk∑j=0n-kn-kjqnkqTn-k-jm-1,-1λ;μ;ν;qTjm-1,11,-x;λ;μ;ν;qznnq!. Comparing coefficients the zn/[n]q!, we obtain (119)Bknx;q=xk∑j=0n-kn-kjqnkqTn-k-jm-1,-1λ;μ;ν;qTjm-1,11,-x;λ;μ;ν;q.Corollary 13.
Forn,j,k∈N0, 0≤k≤j≤n, m∈N, and 0<q<1, one has(120)∑j=0nnjqSj,k;qBn-jk1,-x;q=∑j=0n-kn-kjqnkqTn-k-jm-1,-1λ;μ;ν;qTjm-1,11,-x;λ;μ;ν;q.Proof (see (120)).
Substituting (32) into the left-hand side of (115), we obtain the result.



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