On 𝑞-Operators and Summation of Some 𝑞-Series

Abstract

Using Jackson's 𝑞-derivative and the 𝑞-Stirling numbers, we establish some transformation theorems leading to the values of some convergent 𝑞-series.

1. Introduction

The operator (𝑥(𝑑/𝑑𝑥))^𝑛 has many assets and plays a central role in arithmetic fields and in computation of some finite or infinite sums. For example, when we try to compute the sum ∑^+∞_𝑘=0𝑘^𝑛𝑥^𝑘, we use the operators (𝑥(𝑑/𝑑𝑥))^𝑛, which give^+∞_𝑘=0𝑘^𝑛𝑥^𝑘=𝑥𝑑𝑑𝑥^𝑛1,|𝑥|1−𝑥<1,𝑛=0,1,2…. (1.1) These operators are intimately related to the Stirling numbers of second kind {^𝑛_𝑘} by the formula (see [1])𝑥𝑑𝑑𝑥^𝑛𝑓(𝑥)=^𝑛_𝑘=1𝑛𝑘𝑥^𝑘𝑑^𝑘𝑓𝑑𝑥^𝑘, (1.2) where 𝑓 is a suitable function. We note that the 𝑞-analogue of formula (1.2) has been studied by many authors (see [2, 3] and references therein) and has found applications in many fields such as arithmetic partitions and asymptotic expansions.

This paper deals with the analogues of the operators (𝑥(𝑑/𝑑𝑥))^𝑛 in Quantum Calculus and some 𝑞-transformation theorems that will be used to establish the sums of some 𝑞-series.

This paper is organized as follows. In Section 2, we present some preliminary notions and notations useful in the sequel. Section 3 gives three applications of a result proved in [2], states a transformation theorem using the 𝑞-Stirling numbers, and presents some related applications. Section 4 attempts to give a new 𝑞-analogue of formula (1.2) by studying the transformation theorem related to a 𝑞-derivative operator.

2. Notations and Preliminaries

To make this paper self-containing and easily decipherable, we recall some useful preliminaries about the Quantum Calculus and we select Gasper-Rahman’s book [4], for the notations and for a deep study in this way. Throughout this paper, we fix 𝑞∈]0,1[.

2.1. 𝑞-Shifted Factorials

For 𝑎∈ℂ, the 𝑞-shifted factorials are defined by 𝑎;𝑞₀=1,𝑎;𝑞_𝑛=^𝑛−1_𝑘=01−𝑎𝑞^𝑘,𝑎;𝑞_∞=^+∞_𝑘=01−𝑎𝑞^𝑘. (2.1) We also write 𝑎₁,…,𝑎_𝑘;𝑞_𝑛=^𝑘_𝑗=1𝑎_𝑗;𝑞_𝑛,𝑛=0,1,…,∞. (2.2) We put [𝑥]_𝑞=1−𝑞^𝑥[𝑛]1−𝑞,𝑥∈ℂ,_𝑞!=𝑞;𝑞_𝑛1−𝑞^𝑛,𝑛∈ℕ. (2.3) For 𝑎,𝑥∈ℂ and 𝑛∈ℕ, we adopt the following notation [5]: (𝑥−𝑎)^𝑛_𝑞=1if()⋯𝑛=0,𝑥−𝑎𝑥−𝑎𝑞𝑥−𝑎𝑞^𝑛−1if𝑛⩾1. (2.4) The 𝑞-analogue of the Jordan factorial is given by[𝑥]_𝑘,𝑞=[𝑥]_𝑞[𝑥−1]_𝑞⋯[𝑥−𝑘+1]_𝑞=1−𝑞^𝑥1−𝑞^𝑥−1⋯1−𝑞^{𝑥−𝑘+1}1−𝑞^𝑘, (2.5) and the 𝑞-binomial coefficient is defined by𝑥𝑘_𝑞=[𝑥]_𝑘,𝑞[𝑘]_𝑞!. (2.6)

2.2. The Jackson's 𝑞-Derivative

The 𝑞-derivative 𝐷_𝑞𝑓 of a function 𝑓 is defined by (see [4])𝐷_𝑞𝑓(𝑥)=𝑓𝑞𝑥−𝑓(𝑥)𝑞−1𝑥if𝑥≠0, (2.7) and (𝐷_𝑞𝑓)(0)=𝑓^′(0) provided 𝑓^′(0) exists. Note that when 𝑓 is differentiable, at 𝑥, then (𝐷_𝑞)𝑓(𝑥) tends to 𝑓^′(𝑥) as 𝑞 tends to 1⁻.

It is easy to see that for suitable functions 𝑓 and 𝑔, we have𝐷_𝑞𝑓𝑔(𝑥)=𝑓𝑞𝑥𝐷_𝑞𝑔(𝑥)+𝑔(𝑥)𝐷_𝑞𝑓(𝑥),𝐷_𝑞𝑓𝑔(𝑥)=𝑔𝑞𝑥𝐷_𝑞𝑓(𝑥)−𝑓𝑞𝑥𝐷_𝑞𝑔(𝑥)𝑔(𝑥)𝑔.𝑞𝑥 (2.8)(2.9)

2.3. Elementary 𝑞-Special Functions

Two 𝑞-analogues of the exponential function are given by (see [4]) 𝑒_𝑞(𝑧)=^+∞_𝑛=0𝑧^𝑛[𝑛]_𝑞!=11−𝑞𝑧;𝑞_∞,|𝑧|<1−𝑞⁻¹,𝐸_𝑞(𝑧)=^+∞_𝑛=0𝑞^{𝑛(𝑛−1)/2}𝑧^𝑛[𝑛]_𝑞!=−1−𝑞𝑧;𝑞_∞,𝑧∈ℂ. (2.10) They satisfy the relations 𝐷_𝑞𝑒_𝑞(𝑧)=𝑒_𝑞(𝑧),𝐷_𝑞𝐸_𝑞(𝑧)=𝐸_𝑞,𝑒𝑞𝑧_𝑞(𝑧)𝐸_𝑞(−𝑧)=𝐸_𝑞(𝑧)𝑒_𝑞(−𝑧)=1,𝐸_𝑞(𝑧)=𝑒_1/𝑞(𝑧). (2.11) In 1910, F. H. Jackson defined a 𝑞-analogue of the Gamma function by (see [4, 6]) Γ_𝑞(𝑥)=𝑞;𝑞_∞𝑞^𝑥;𝑞_∞1−𝑞^1−𝑥,𝑥≠0,−1,−2,…. (2.12) It satisfies the following functional equations: Γ_𝑞(𝑥+1)=[𝑥]_𝑞Γ_𝑞(𝑥),Γ_𝑞(1)=1,Γ_𝑞(𝑛+1)=[𝑛]_𝑞!,𝑛∈ℕ. (2.13)

2.4. 𝑞-Stirling Numbers of Noncentral Type

In [7], Charalambides introduced the so-called noncentral 𝑞-Stirling numbers, which are 𝑞-analogues of the Stirling numbers and classified into two kinds.

The noncentral 𝑞-Stirling numbers of the first kind 𝑠_𝑞(𝑛,𝑘;𝑟) are defined by the following generating relation:[𝑡−𝑟]_𝑛,𝑞=𝑞^{−𝑛2𝑛−𝑟𝑛}_𝑘=0𝑠_𝑞(𝑛,𝑘;𝑟)[𝑡]^𝑘_𝑞,𝑛=0,1,…, (2.14) and they are given by𝑠_𝑞()=1𝑛,𝑘;𝑟1−𝑞^{𝑛𝑛−𝑘}_𝑗=𝑘(−1)^𝑗−𝑘𝑞^{2𝑛−𝑗+𝑟(𝑛−𝑗)}𝑛𝑗_𝑞𝑗𝑘. (2.15) The noncentral 𝑞-Stirling numbers of the second kind 𝑆_𝑞(𝑛,𝑘;𝑟) are defined by the following generating relation:[𝑡]^𝑛_𝑞=^𝑛_𝑘=0𝑞^{𝑘2−𝑟𝑘}𝑆_𝑞(𝑛,𝑘;𝑟)[𝑡−𝑟]_𝑘,𝑞,𝑛=0,1,…, (2.16) and they are given by𝑆_𝑞()=1𝑛,𝑘;𝑟[𝑘]_𝑞!^𝑘_𝑗=0(−1)^𝑘−𝑗𝑞^{2𝑗+1−(𝑟+𝑗)𝑘}𝑘𝑗_𝑞𝑟+𝑗^𝑛_𝑞=11−𝑞^{𝑛𝑛−𝑘}_𝑗=𝑘(−1)^𝑗−𝑘𝑞^{𝑟(𝑗−𝑘)}𝑛𝑗𝑗𝑘_𝑞. (2.17)

Remark 2.1. Note that when 𝑟=0, then 𝑠_𝑞(𝑛,𝑘;𝑟) and 𝑆_𝑞(𝑛,𝑘;𝑟) reduce to the 𝑞-Stirling numbers, respectively, of the first and the second kind studied by Gould, Carlitz, and Kim (see [8–11]).

Properties
The noncentral 𝑞-Stirling numbers satisfy the following properties. (i)For 𝑛=1,2,… and 𝑘=1,2,…,𝑛, 𝑠_𝑞(𝑛,𝑘;𝑟)=𝑠_𝑞(𝑛−1,𝑘−1;𝑟)−[𝑛+𝑟−1]_𝑞𝑠_𝑞(𝑛−1,𝑘;𝑟), (2.18) under the following conditions: 𝑠_𝑞(0,0;𝑟)=1,𝑠_𝑞(𝑛,0;𝑟)=𝑞^{𝑛2+𝑟𝑛}[−𝑟]_𝑛,𝑞𝑠,𝑛>0,_𝑞(0,𝑘;𝑟)=0,𝑘>0,𝑠_𝑞(𝑛,𝑘;𝑟)=0,𝑘>𝑛. (2.19)(ii)For 𝑛=1,2,… and 𝑘=1,2,…,𝑛, 𝑆_𝑞(𝑛,𝑘;𝑟)=𝑆_𝑞(𝑛−1,𝑘−1;𝑟)+[𝑟+𝑘]_𝑞𝑆_𝑞(𝑛−1,𝑘;𝑟), (2.20) under the following conditions: 𝑆_𝑞(0,0;𝑟)=1,𝑆_𝑞(𝑛,0;𝑟)=[𝑟]^𝑛_𝑞𝑆,𝑛>0,_𝑞(0,𝑘;𝑟)=0,𝑘>0,𝑆_𝑞(𝑛,𝑘;𝑟)=0,𝑘>𝑛. (2.21)

3. The Operator (𝑥𝐷_𝑞)^𝑚 and Some Related Transformations Theorems

As in the classical case (see [1]), the iterate (𝑥𝐷_𝑞)^𝑚, 𝑚∈ℕ, can be expanded in finite terms involving the 𝑞-Stirling numbers. This is the purpose of the following result.

Lemma 3.1 (see [2, 3]). Letting 𝑓 be a differentiable function, then one has 𝑥𝐷_𝑞^𝑚𝑓(𝑥)=^𝑚_𝑘=1𝑚𝑘_𝑞,1𝑥^𝑘𝐷^𝑘_𝑞𝑓(𝑥),𝑚=1,2,…, (3.1) where 𝑚𝑘_𝑞,1=𝑞^{𝑘(𝑘−1)/2}𝑆_𝑞()=1𝑚−1,𝑘−1;1[𝑘−1]_𝑞!^𝑘−1_𝑗=0(−1)^𝑗𝑞^𝑗2𝑗𝑘−1_𝑞𝑘−𝑗_𝑞^𝑚−1. (3.2)

Now, let us give three applications of the previous lemma.

Example 3.2 (𝑞-binomial series). The 𝑞-binomial theorem asserts that ₁Φ₀𝑞^𝑎;−;𝑞,𝑥=^+∞_𝑛=0𝑞^𝑎;𝑞_𝑛𝑞;𝑞_𝑛𝑥^𝑛=𝑞^𝑎𝑥;𝑞_∞𝑥;𝑞_∞,|𝑥|<1. (3.3) Using the fact that, for all 𝑚∈ℕ, 𝑥𝐷_𝑞^𝑚𝑥^𝑛=[𝑛]^𝑚_𝑞𝑥^𝑛 (3.4) and the previous lemma, we deduce that ^+∞_𝑛=0𝑞^𝑎;𝑞_𝑛𝑞;𝑞_𝑛[𝑛]^𝑚_𝑞𝑥^𝑛=^𝑚_𝑘=1𝑚𝑘_𝑞,1𝑥^𝑘𝐷^𝑘_𝑞𝑞^𝑎𝑥;𝑞_∞𝑥;𝑞_∞,|𝑥|<1. (3.5) On the other hand, the definition of 𝑞-derivative (2.9) gives 𝐷_𝑞𝑞^𝑎𝑥;𝑞_∞𝑥;𝑞_∞=[𝑎]_𝑞𝑞^𝑎+1𝑥;𝑞_∞𝑥;𝑞_∞, (3.6) and by iteration we have 𝐷^𝑘_𝑞𝑞^𝑎𝑥;𝑞_∞𝑥;𝑞_∞=Γ_𝑞(𝑎+𝑘)Γ_𝑞(𝑎)𝑞^𝑎+𝑘𝑥;𝑞_∞𝑥;𝑞_∞. (3.7) Thus, ^+∞_𝑛=0𝑞^𝑎;𝑞_𝑛𝑞;𝑞_𝑛[𝑛]^𝑚_𝑞𝑥^𝑛=1Γ_𝑞(𝑎)𝑥;𝑞_∞^𝑚_𝑘=1𝑚𝑘_𝑞,1𝑥^𝑘Γ_𝑞(𝑎+𝑘)𝑞^𝑎+𝑘𝑥;𝑞_∞. (3.8) So, taking 𝑎=1, we obtain ^+∞_𝑛=0[𝑛]^𝑚_𝑞𝑥^𝑛=11−𝑥^𝑚_𝑘=1𝑚𝑘_𝑞,1𝑥^𝑘[𝑘]_𝑞!𝑥𝑞;𝑞_𝑘. (3.9) Remark that if 𝑞 tends to 1⁻, we obtain the formula given in [1, page 366].

Example 3.3 (𝑞-Bessel function). We consider the function 𝐶_𝑝(𝑥)=^+∞_𝑘=0(−1)^𝑘𝑥^𝑘[𝑘]_𝑞!𝑘+𝑝_𝑞!=𝑥2^−𝑝𝐽_𝑝⁽¹⁾1−𝑞√2𝑥,𝑞, (3.10) where 𝐽_𝑝⁽¹⁾(⋅,𝑞) is the first Jackson's 𝑞-Bessel function of order 𝑝 (see [12, 13]).
By application of the operator (𝑥𝐷_𝑞)^𝑚 to 𝐶₀(𝑥) and the use of relation (3.4), we obtain 𝑥𝐷_𝑞^𝑚𝐶₀(𝑥)=^+∞_𝑘=0(−1)^𝑘[𝑘]^𝑚_𝑞[𝑘]_𝑞!²𝑥^𝑘. (3.11) Then, using Lemma 3.1 and the fact that 𝐷_𝑞𝐶_𝑝(𝑥)=−𝐶_𝑝+1(𝑥), (3.12) we get ^+∞_𝑘=0(−1)^𝑘[𝑘]^𝑚_𝑞[𝑘]_𝑞!²𝑥^𝑘=^𝑚_𝑘=1𝑚𝑘_𝑞,1(−1)^𝑘𝑥^𝑘𝐶_𝑘(𝑥). (3.13)

Example 3.4 (𝑞-polynomial exponential). Take 𝑓(𝑥)=𝑒_𝑞(𝑥). From relation (3.4) and Lemma 3.1, we obtain 𝑥𝐷_𝑞^𝑚𝑒_𝑞(𝑥)=𝑒_𝑞(𝑥)Φ_𝑚,𝑞(𝑥)=^+∞_𝑛=1[𝑛]^𝑚_𝑞[𝑛]_𝑞!𝑥^𝑛, (3.14) where Φ_𝑚,𝑞(𝑥)=^𝑚_𝑘=1𝑚𝑘_𝑞,1𝑥^𝑘, (3.15) which is called the 𝑞-polynomial exponential. So, ^𝑚_𝑘=1𝑚𝑘_𝑞,1𝑥^𝑘=𝐸_𝑞(−𝑥)^+∞_𝑛=1[𝑛]^𝑚_𝑞[𝑛]_𝑞!𝑥^𝑛=^+∞𝑛_𝑛=1𝑘=1(−1)^𝑘𝑞^𝑘2[𝑛−𝑘]^𝑚_𝑞[𝑘]_𝑞![𝑛−𝑘]_𝑞!𝑥^𝑛. (3.16)
In many mathematical fields there are some transformation theorems using the Stirling numbers leading one to compute certain sums (see [14]). The purpose of the following result is to give a 𝑞-analogue context.

Theorem 3.5. Let 𝑓(𝑥) and 𝑔(𝑥) be two functions satisfying 𝑓(𝑥)=^+∞_𝑛=0𝑎_𝑛[𝑥]^𝑛_𝑞,𝑔(𝑥)=^+∞_𝑛=0𝑐_𝑛𝑥^𝑛. (3.17) Then ^+∞_𝑛=0𝑐_𝑛𝑓(𝑛)𝑥^𝑛=^+∞_𝑚=1𝑎_𝑚^𝑚_𝑘=1𝑚𝑘_𝑞,1𝑥^𝑘𝐷_𝑞^𝑘𝑔(𝑥) (3.18) provided the series ^+∞_𝑛=0𝑐_𝑛𝑓(𝑛)𝑥^𝑛 (3.19) converges absolutely.

Proof. From Lemma 3.1 and the properties of the 𝑞-Stirling numbers of the second kind (2.21), we obtain ^+∞_𝑚=1𝑎_𝑚^𝑚_𝑘=1𝑚𝑘_𝑞,1𝑥^𝑘𝐷^𝑘_𝑞𝑔(𝑥)=^+∞_𝑚=1𝑎_𝑚𝑥𝐷_𝑞^𝑚𝑔(𝑥)=^+∞_𝑚=1𝑎_𝑚𝑥𝐷_𝑞^𝑚+∞_𝑛=0𝑐_𝑛𝑥^𝑛. (3.20) The result follows, then, from relations (3.4) and (3.19).

Corollary 3.6. Let 𝑓(𝑥)=∑^+∞_𝑛=0𝑎_𝑛[𝑥]^𝑛_𝑞. Then ^+∞_𝑛=0𝑓(𝑛)𝑥^𝑛=^+∞_𝑚=1𝑎_𝑚^𝑚_𝑘=1𝑚𝑘_𝑞,1𝑥^𝑘[𝑘]_𝑞!𝑥;𝑞_𝑘+1 (3.21) provided the series ∑^+∞_𝑛=0𝑓(𝑛)𝑥^𝑛 converges absolutely.

Proof. By taking 𝑔(𝑥)=1/(1−𝑥)=∑^+∞_𝑛=0𝑥^𝑛, |𝑥|<1, in the previous theorem, and by application of relation (3.7), we obtain ^+∞_𝑛=0𝑓(𝑛)𝑥^𝑛=^+∞_𝑚=1𝑎_𝑚^𝑚_𝑘=1𝑚𝑘_𝑞,1𝑥^𝑘𝐷^𝑘_𝑞1=1−𝑥_+∞𝑚=1𝑎_𝑚^𝑚_𝑘=1𝑚𝑘_𝑞,1𝑥^𝑘Γ_𝑞(𝑘+1)Γ_𝑞(1)𝑞^𝑘+1𝑥;𝑞_∞𝑥;𝑞_∞=^+∞_𝑚=1𝑎_𝑚^𝑚_𝑘=1𝑚𝑘_𝑞,1𝑥^𝑘[𝑘]_𝑞!𝑥;𝑞_𝑘+1. (3.22)

Example 3.7. Let 𝑓(𝑥)=[𝑥]_𝑛,𝑞. Then, the fact that [𝑥]_𝑛,𝑞=𝑞^−𝑛2𝑛_𝑘=0𝑠_𝑞(𝑛,𝑘;0)[𝑥]^𝑘_𝑞,𝑛=0,1,… (3.23) and Corollary 3.6 give ^+∞_𝑘=0[𝑘]_𝑛,𝑞𝑥^𝑘=𝑞^−𝑛21−𝑥^𝑛_𝑚=1𝑠_𝑞(𝑛,𝑚;0)^𝑚_𝑙=1𝑚𝑙_𝑞,1𝑥^𝑙[𝑙]_𝑞!𝑞𝑥;𝑞_𝑙. (3.24) Remark that when 𝑞 tends to 1⁻, we obtain the formula given in [14, (6.4), page 3863].

Some others summation formulas are presented in the following statements.

Corollary 3.8. For 𝑓(𝑥)=∑^+∞_𝑛=0𝑎_𝑛[𝑥]^𝑛_𝑞, the transformation formulas lead to the following: (1)∑^+∞_𝑛=0𝑞^{𝑛(𝑛−1)/2}[^𝛼_𝑛]_𝑞𝑓(𝑛)𝑥^𝑛=∑^+∞_𝑛=1𝑎_𝑛∑^𝑛_𝑘=1𝑞^{𝑘(𝑘−1)/2}{^𝑛_𝑘}_𝑞,1[𝑘]_𝑞![^𝛼_𝑘]_𝑞𝑥^𝑘(1+𝑞^𝑘𝑥)_𝑞^𝛼−𝑘; (2)∑^+∞_𝑛=0((1−𝑞^𝛼)^𝑛_𝑞/(1−𝑞)^𝑛_𝑞)𝑓(𝑛)𝑥^𝑛=∑^+∞_𝑛=1𝑎_𝑛∑^𝑛_𝑘=1{^𝑛_𝑘}_𝑞,1^{𝛼+𝑘−1}_𝑘−1𝑞(𝑥^𝑘/(1−𝑥)_𝑞^𝛼+𝑘); (3)∑^+∞_𝑛=0(𝑓(𝑛)/[𝑛]_𝑞!)𝑥^𝑛=𝑒_𝑞(𝑥)∑^+∞_𝑛=1𝑎_𝑛∑^𝑛_𝑘=1{^𝑛_𝑘}_𝑞,1𝑥^𝑘=𝑒_𝑞(𝑥)∑^+∞_𝑛=1𝑎_𝑛Φ_𝑛,𝑞(𝑥); (4)∑^+∞_𝑛=0𝑞^{𝑛(𝑛−1)}(𝑓(𝑛)/[𝑛]_𝑞!)𝑥^𝑛=∑^+∞_𝑛=1𝑎_𝑛∑^𝑛_𝑘=1{^𝑛_𝑘}_𝑞,1𝑞^{𝑘(𝑘−1)/2}(𝑥^𝑘/(−(1−𝑞)𝑞^𝑘𝑥;𝑞)_∞) provided the series converge absolutely.

Proof. The results are direct consequences of Theorem 3.5 by putting the following: (1)𝑔(𝑥)=(1+𝑥)^𝛼_𝑞=∑^+∞_𝑛=0𝑞^{𝑛(𝑛−1)/2}[^𝛼_𝑛]_𝑞𝑥^𝑛 and remark that 𝐷^𝑘_𝑞(1+𝑥)^𝛼_𝑞=𝑞^{𝑘(𝑘−1)/2}[𝛼]_𝑞[𝛼−1]_𝑞⋯[𝛼−𝑘+1]_𝑞(1+𝑞^𝑘𝑥)_𝑞^𝛼−𝑘;(2)𝑔(𝑥)=1/(1−𝑥)^𝛼_𝑞=∑^+∞_𝑛=0((1−𝑞^𝛼)^𝑛_𝑞/(1−𝑞)^𝑛_𝑞)𝑥^𝑛 and remark that 𝐷^𝑘_𝑞(1/(1−𝑥)^𝛼_𝑞)=[𝛼]_𝑞[𝛼+1]_𝑞⋯[𝛼+𝑘−1]_𝑞/(1−𝑥)_𝑞^𝛼+𝑘;(3)𝑔(𝑥)=𝑒_𝑞(𝑥); (4)𝑔(𝑥)=𝐸_𝑞(𝑥) and remark that 𝐷^𝑘_𝑞𝐸_𝑞(𝑥)=𝑞^{𝑘(𝑘−1)/2}𝐸_𝑞(𝑞^𝑘𝑥).

Remark 3.9. The last formulas coincide with some of the ones given in [15] when 𝑞 tends to 1⁻.

4. The Operator ((𝑥;𝑞)₁𝐷_𝑞)^𝑚 and Related Transformation Theorem

Lemma 4.1. For a suitable function 𝑓, one has for 𝑚=1,2,…(𝑥;𝑞)₁𝐷_𝑞^𝑚𝑓(𝑥)=^𝑚_𝑘=1(−1)^𝑚−𝑘𝑆_𝑞(𝑚−1,𝑘−1;1)𝑥;𝑞_𝑘𝐷^𝑘_𝑞𝑓(𝑥). (4.1)

Proof. The formula can be obtained by induction with respect to 𝑚. Indeed, for 𝑚=1, we have 𝑥;𝑞₁𝐷_𝑞𝑓(𝑥)=𝑥;𝑞₁𝐷_𝑞𝑓(𝑥)=𝑆_𝑞(0,0;1)𝑥;𝑞₁𝐷_𝑞𝑓(𝑥). (4.2) Assuming that formula (4.1) is true for 𝑚, then 𝑥;𝑞₁𝐷_𝑞^𝑚+1𝑓(𝑥)=𝑥;𝑞₁𝐷_𝑞^𝑚_𝑘=1(−1)^𝑚−𝑘𝑆_𝑞(𝑚−1,𝑘−1;1)𝑥;𝑞_𝑘𝐷^𝑘_𝑞𝑓(𝑥=)𝑥;𝑞₁^𝑚_𝑘=1(−1)^𝑚−𝑘𝑆_𝑞(𝑚−1,𝑘−1;1)𝑞𝑥;𝑞_𝑘𝐷_𝑞^𝑘+1𝑓(𝑥)−𝑥;𝑞₁^𝑚_𝑘=1(−1)^𝑚−𝑘𝑆_𝑞(𝑚−1,𝑘−1;1)[𝑘]_𝑞𝑞𝑥;𝑞_𝑘−1𝐷^𝑘_𝑞𝑓(𝑥)=^𝑚+1_𝑘=2(−1)^{𝑚−𝑘+1}𝑆_𝑞(𝑚−1,𝑘−2;1)𝑥;𝑞_𝑘𝐷^𝑘_𝑞𝑓(𝑥)−^𝑚_𝑘=1(−1)^𝑚−𝑘[𝑘]_𝑞𝑆_𝑞(𝑚−1,𝑘−1;1)𝑥;𝑞_𝑘𝐷^𝑘_𝑞𝑓(𝑥)=^𝑚_𝑘=2(−1)^{𝑚−𝑘+1}𝑆_𝑞(𝑚−1,𝑘−2;1)−[𝑘]_𝑞𝑆_𝑞(𝑚−1,𝑘−1;1)𝑥;𝑞_𝑘𝐷^𝑘_𝑞𝑓(𝑥)+𝑆_𝑞(𝑚−1,𝑚−1;1)𝑥;𝑞_𝑚+1𝐷_𝑞^𝑚+1𝑓(𝑥)−(−1)^𝑚−1𝑆_𝑞(𝑚−1,0;1)𝑥;𝑞₁𝐷_𝑞𝑓(𝑥). (4.3) The result is easily deduced by formulas (2.20), and (2.21).

Theorem 4.2. Let 𝑓(𝑥) and 𝑔(𝑥) be two functions defined by 𝑓(𝑥)=^+∞_𝑛=0(−1)^𝑛𝛼_𝑛[𝑥]^𝑛_𝑞,𝑔(𝑥)=^+∞_𝑛=0𝑐_𝑛𝑥;𝑞_𝑛. (4.4) If the series ^+∞_𝑛=0𝑐_𝑛𝑓(𝑛)𝑥;𝑞_𝑛 (4.5) converges absolutely, then ^+∞_𝑛=0𝑐_𝑛𝑓(𝑛)𝑥;𝑞_𝑛=^+∞_𝑚=1𝛼_𝑚^𝑚_𝑘=1(−1)^𝑚−𝑘𝑆_𝑞(𝑚−1,𝑘−1;1)𝑥;𝑞_𝑘𝐷^𝑘_𝑞𝑔(𝑥). (4.6)

Proof. From the previous lemma, we obtain for 𝑚=1,2,…, ^+∞_𝑚=0𝛼_𝑚(𝑥;𝑞)₁𝐷_𝑞^𝑚𝑔(𝑥)=^+∞_𝑚=1𝛼_𝑚^𝑚_𝑘=1(−1)^𝑚−𝑘𝑆_𝑞(𝑚−1,𝑘−1;1)𝑥;𝑞_𝑘𝐷^𝑘_𝑞𝑔(𝑥). (4.7) So, the absolute convergence of the series (4.5) and the fact that (𝑥;𝑞)₁𝐷_𝑞^𝑚𝑥;𝑞_𝑛=(−1)^𝑚[𝑛]^𝑚_𝑞𝑥;𝑞_𝑛,𝑚∈ℕ (4.8) achieve the proof.

Corollary 4.3. Let 𝑓(𝑥)=∑^+∞_𝑚=0(−1)^𝑚𝛼_𝑚[𝑥]^𝑚_𝑞. Then ^+∞_𝑚=1𝛼_𝑚^𝑚_𝑘=1(−1)^𝑚−𝑘𝑆_𝑞(𝑚−1,𝑘−1;1)𝑥;𝑞_𝑘[𝑛]_𝑘,𝑞𝑥^𝑛−𝑘=^𝑛_𝑘=0−𝑞^𝑘𝑠_𝑞(𝑘,0,−𝑛)𝑓(𝑘)𝑥;𝑞_𝑘. (4.9)

Proof. Put 𝑔(𝑥)=𝑥^𝑛.
Using the representation (see [5]) 𝑥^𝑛=^𝑛_𝑘=0(−1)^𝑘𝑛𝑘_𝑞𝑞^{−(𝑛−1)𝑘}𝑞^𝑘2𝑥;𝑞_𝑘=^𝑛_𝑘=0−𝑞^𝑘𝑠_𝑞(𝑘,0,−𝑛)𝑥;𝑞_𝑘, (4.10) relation (4.6) and the fact that 𝐷^𝑘_𝑞𝑥^𝑛=[𝑛]_𝑘,𝑞𝑥^𝑛−𝑘 (4.11) give the desired result.

Remark 4.4. Note that recently Liu in his paper (see [16]) has obtained some interesting 𝑞-identities in showing that the solutions of two difference equations involve some series of 𝑞-operators 𝐷^𝑛_𝑞 of 𝑞-Cauchy type.

Acknowledgments

The authors would like to thank the Board of editors for their helpful comments. They are thankful to the anonymous reviewer for his remarks, who pointed the references out to them (see [3, 16]).