𝑄-Analogues of Symbolic Operators

Abstract

Here presented are 𝑞-extensions of several linear operators including a novel 𝑞-analogue of the derivative operator 𝐷. Some 𝑞-analogues of the symbolic substitution rules given by He et al., 2007, are obtained. As sample applications, we show how these 𝑞-substitution rules may be used to construct symbolic summation and series transformation formulas, including 𝑞-analogues of the classical Euler transformations for accelerating the convergence of alternating series.

1. Definitions and Basic Identities

Unless otherwise stated, we consider all operators to act on formal power series in the single variable 𝑡, with coefficients possibly depending on 𝑞. We assume that 0<|𝑞|<1. Issues of convergence will be addressed in a later paper.

We will use 1 to denote the identity operator and define the following operators: (1)𝐸_𝑞𝑓(𝑡)=𝑓(𝑡𝑞) (forward multiplicative shift), (2)Δ_𝑞𝑓(𝑡)=𝑓(𝑡𝑞)−𝑓(𝑡) (forward 𝑞-difference), (3)𝐿_𝑞𝑓(𝑡)=𝑡(log𝑞)𝑓^′(𝑡) (forward logarithmic shift).

The first two of these can be regarded as 𝑞-analogues of the ordinary (additive) shift and forward difference operators, respectively. 𝐿_𝑞 will play a role similar to that of the derivative 𝐷.

The operator inverse of 𝐸_𝑞 (which we denote as 𝐸_𝑞⁻¹) clearly exists and is equal to 𝐸_𝑞−1. We define the central 𝑞-difference operator 𝛿_𝑞 by 𝛿_𝑞=𝑓𝑡𝑞^1/2−𝑓𝑡𝑞^−1/2, (1) and note that 𝛿_𝑞=Δ_𝑞𝐸_𝑞^−1/2=Δ_𝑞/𝐸_𝑞^1/2,𝛿_𝑞^2𝑘=Δ_𝑞^2𝑘𝐸_𝑞^−𝑘.

The previous 𝑞-operators are linear and satisfy some familiar identities, for example, 𝐸_𝑞=1+Δ_𝑞. The binomial identity Δ^𝑛_𝑞=^𝑛_𝑘=0(−1)^𝑛−𝑘𝑛𝑘𝐸^𝑘_𝑞 (2) can be established by induction or by considering the operator expansion of (𝐸_𝑞−1)^𝑛.

Treating these operators formally, we need only to consider their effect on nonnegative integer powers of 𝑡.  𝐸_𝑞,  Δ_𝑞, and 𝐿_𝑞 are “diagonal” in the sense that each maps 𝑡^𝑘↦𝑀(𝑞,𝑘)𝑡^𝑘, with the function 𝑀 depending on the particular operator. For example, Δ_𝑞[𝑡^𝑘]=(𝑞^𝑘−1)𝑡^𝑘 for 𝑘>0, and Δ_𝑞[1]=0. Similarly, 𝐿_𝑞[𝑡^𝑘]=𝑡^𝑘log(𝑞^𝑘).

With this observation, it is easy to verify many additional identities. For example, consider the alternating geometric series ∑^∞_𝑛=0(−1)^𝑛Δ^𝑛_𝑞 applied to 𝑡^𝑘. We have ^∞_𝑛=0(−1)^𝑛Δ^𝑛_𝑞𝑡^𝑘=𝑡^𝑘∞_𝑛=0(−1)^𝑛𝑞^𝑘−1^𝑛=𝑡^𝑘11−1−𝑞^𝑘=𝑡^𝑘𝑞^−𝑘. (3) In other words, this formal power series gives the operator 𝐸_𝑞−1. Stated differently, 1+Δ_𝑞⁻¹=𝐸_𝑞⁻¹=𝐸_𝑞−1=^∞_𝑛=0(−1)^𝑛Δ^𝑛_𝑞, (4) which is exactly the result we should expect. We may establish the following identities in similar fashion: 1−Δ_𝑞⁻¹=^∞_𝑛=0Δ^𝑛_𝑞,log1+Δ_𝑞=^∞_𝑛=1(−1)^𝑛+1𝑛Δ^𝑛_𝑞=𝐿_𝑞,𝑒^𝐿𝑞=^∞_𝑛=01𝐿𝑛!^𝑛𝑞=𝐸_𝑞. (5)(6)(7) In addition to these last two identities, 𝐿_𝑞 obeys the product rule 𝐿_𝑞𝑓(𝑡)𝑔(𝑡)=𝐿_𝑞𝑓(𝑡)𝑔(𝑡)+𝑓(𝑡)𝐿_𝑞𝑔(𝑡), (8) so that 𝐿_𝑞 is a 𝑞-analogue of the ordinary derivative operator 𝐷.

2. Main Results

We begin with some 𝑞-analogues of the symbolic substitution rules in [1] (specifically, (2.4) and (2.5)).

Proposition 1. Let 𝐹(𝑡) have the formal power series expansion 𝐹(𝑡)=∑_𝑘≥0𝑓_𝑘𝑡^𝑘, with coefficients possibly dependent on 𝑞. One may obtain operational formulas according to the following rules.(1)The substitution𝑡↦𝐸_𝑞leads to the symbolic formula𝐹𝐸_𝑞=^∞_𝑘=0𝑓_𝑘𝐸^𝑘_𝑞. (9)(2)If𝐹(𝑡)=𝐺(𝑡,𝑒^𝑡), the substitution𝑡↦𝐿_𝑞leads to𝐺𝐿_𝑞,𝐸_𝑞=^∞_𝑘=0𝑓_𝑘𝐿^𝑘_𝑞. (10)(3)If𝐹(𝑡)=𝐺(𝑡,log(1+𝑡)), the substitution𝑡↦Δ_𝑞leads to𝐺Δ_𝑞,𝐿_𝑞=^∞_𝑘=0𝑓_𝑘Δ^𝑘_𝑞. (11)

Note that each of the identities in (5)–(7) can be obtained from elementary Maclaurin series by applying one of these substitution rules. We now present a less trivial example.

For 𝑘 a positive integer, let 𝛼_𝑘(𝑥) denote the Eulerian fraction (cf. [2], p. 245). It is well known that ^∞_𝑗=0𝑗^𝑘𝑥^𝑗=𝐴_𝑘(𝑥)(1−𝑥)^𝑘+1=𝛼_𝑘(𝑥),(|𝑥|<1), (12) where 𝐴_𝑘(𝑥) is the 𝑘th Eulerian polynomial. Additionally, ([3], p. 24) gives the formula 1−𝑥𝑒^𝑡−1=^∞_𝑘=0𝛼_𝑘(𝑥)𝑡^𝑘.𝑘! (13) Substituting 𝑡↦𝐿_𝑞 leads to the formal identity 1−𝑥𝐸_𝑞⁻¹=^∞_𝑘=0𝛼_𝑘(𝑥)𝐿𝑘!^𝑘𝑞. (14)

We can obtain additional identities in this fashion from other expansions of (1−𝑥𝑒^𝑡)⁻¹. For example, if 𝑥≠0 and 𝑥≠1, we have the following analogues of (3.1)–(3.4) in [4]: 1−𝑥𝐸_𝑞⁻¹=^∞_𝑘=0𝑥^𝑘(1−𝑥)^𝑘+1Δ^𝑘_𝑞,1−𝑥𝐸_𝑞⁻¹=^∞_𝑘=0𝑥(1−𝑥)^2𝑘+1𝑥⁻¹Δ_𝑞^2𝑘𝐸^𝑘_𝑞−Δ_𝑞^2𝑘𝐸_𝑞^𝑘+1,1−𝑥𝐸_𝑞⁻¹=1+^∞_𝑘=0𝑥(1−𝑥)^2𝑘+1Δ_𝑞^2𝑘𝐸_𝑞^𝑘−1Δ−𝑥^𝑞2𝑘𝐸^𝑘_𝑞. (15)(16)(17) Direct proofs of (14)–(17) are given in Section 5.

Proposition 2. For a given function 𝑓(𝑡), define 𝐹_𝑞(𝑥)=∑_𝑘≥0𝑓(𝑞^𝑘)𝑥^𝑘. If 𝑥≠0 and 𝑥≠1, 𝐹_𝑞(𝑥)=^∞_𝑘=0𝛼_𝑘(𝑥)𝐿𝑘!^𝑘𝑞𝑓(1),𝐹_𝑞(𝑥)=^∞_𝑘=0𝑥^𝑘(1−𝑥)^𝑘+1Δ^𝑘_𝑞𝑓(1),𝐹_𝑞(𝑥)=^∞_𝑘=0𝑥(1−𝑥)^2𝑘+1𝑥⁻¹𝛿_𝑞^2𝑘𝑓(1)−𝛿_𝑞^2𝑘𝑓𝑞⁻¹,𝐹_𝑞(𝑥)=1+^∞_𝑘=0𝑥(1−𝑥)^2𝑘+1𝛿_𝑞^2𝑘𝑓𝑞−𝑥𝛿_𝑞^2𝑘𝑓(1). (18)(19)(20)(21)

Proof. Clearly, these follow by applying the operators in (14)–(17) to the function 𝑓(𝑡) and then evaluating at 𝑡=1.

3. Some Applications

As an application, taking 𝑓(𝑡)=1/(log_𝑞(𝑡)+1),  𝑥=−1 in (19) leads to _𝑘≥0(−1)^𝑘1𝑘+1=_𝑘≥0(−1)^𝑘2^𝑘𝑘+1_𝑗=0𝑘𝑗(−1)^𝑘−𝑗1=𝑗+1_∞𝑘=01(𝑘+1)2^𝑘𝑘+1_𝑗=0(−1)^𝑗=𝑘+1𝑗+1_∞𝑘=01(𝑘+1)2^𝑘+1=^∞_𝑘=11𝑘2^𝑘, (22) which gives 1ln2=2+12⋅2²+13⋅2³+14⋅2⁴+⋯. (23) The rate of convergence of this series is 𝑂(1/2^𝑛), much faster than ln2=_𝑘≥0(−1)^𝑘1,𝑘+1 (24) whose convergence rate is 𝑂(1/𝑛).

As for a second application, we may substitute 𝑥=−1 in Proposition 2, obtaining the following series transformation formulas: _𝑘≥0(−1)^𝑘𝑓𝑞^𝑘=^∞_𝑘=0𝛼_𝑘(−1)𝐿𝑘!^𝑘𝑞𝑓(1),_𝑘≥0(−1)^𝑘𝑓𝑞^𝑘=^∞_𝑘=0(−1)^𝑘2^𝑘+1Δ^𝑘_𝑞𝑓(1),_𝑘≥0(−1)^𝑘𝑓𝑞^𝑘=^∞_𝑘=0(−1)^𝑘4^𝑘+1𝛿_𝑞^2𝑘𝑓(1)+𝛿_𝑞^2𝑘𝑓𝑞⁻¹,_𝑘≥1(−1)^𝑘𝑓𝑞^𝑘=1+^∞_𝑘=0−14^𝑘+1𝛿_𝑞^2𝑘𝑓𝑞+𝛿_𝑞^2𝑘𝑓(1). (25) These four identities appear to be novel and could be used to accelerate slowly convergent alternating series ∑^∞_𝑘=1(−1)^𝑘𝑓(𝑞^𝑘). We consider them as 𝑞-analogues of the ordinary Euler transformations.

4. Extensions of the Main Results

All operational formulas presented in Proposition 1 can be extended and the corresponding symbolic substitution formulas established accordingly with an analogous form of (10). For example, we may consider a generating function of the form _𝑘≥0𝑓_𝑘𝑡^𝑘=𝐹𝑡,𝑒^𝑡,𝑒^𝛼𝑡. (26) Letting 𝑡↦𝐿_𝑞 gives _𝑘≥0𝑓_𝑘𝐿^𝑘_𝑞=𝐹𝐿_𝑞,𝐸_𝑞,𝐸^𝛼_𝑞. (27) Applying this to the well-known identity _𝑘≥04^𝑘()!𝐵2𝑘^2𝑘𝑡^2𝑘𝑒=𝑡coth𝑡=𝑡^𝑡+𝑒^−𝑡𝑒^𝑡−𝑒^−𝑡, (28) with 𝐵_𝑛 being the 𝑛th Bernoulli number, we obtain _𝑘≥04^𝑘()!𝐵2𝑘^2𝑘𝐿_𝑞^2𝑘=𝐿_𝑞𝐸_𝑞+𝐸_𝑞⁻¹𝐸_𝑞−𝐸_𝑞⁻¹. (29) Hence, we obtain a symbolic formula _𝑘≥04^𝑘()!𝐵2𝑘^2𝑘𝐿_𝑞^2𝑘−1𝐸_𝑞−𝐸_𝑞⁻¹=𝐸_𝑞+𝐸_𝑞⁻¹. (30) Applying this to an infinitely differentiable function 𝑓(𝑡) at 𝑡=1 yields _𝑘≥04^𝑘()!𝐵2𝑘^2𝑘𝐿_𝑞^2𝑘−1𝐸_𝑞−𝐸_𝑞⁻¹𝑓(1)=𝐸_𝑞+𝐸⁻¹_q𝑓(1). (31) Similarly, using the symbolic relation 𝐿_𝑞𝐸_𝑞+𝐸_𝑞⁻¹𝐸_𝑞−𝐸_𝑞⁻¹=𝐿_𝑞1+Δ_𝑞⁻¹−𝐸_𝑞+1⁻¹, (32) we obtain another operational formula −1+_𝑘≥04^𝑘()!𝐵2𝑘^2𝑘𝐿_𝑞^2𝑘−1+𝐸_𝑞+1⁻¹=Δ_𝑞⁻¹, (33) from which one may construct a series transformation formula.

Another extension is a 𝑞-analogue of the symbolic formulas presented in [5], which is actually a Newton series type extension of the symbolic expansions given in [1]. Consider 1+𝐸_𝑞^𝑥𝑓(1)=_𝑘≥0𝑥𝑘𝐸^𝑘_𝑞𝑓(1)=_𝑘≥0𝑓𝑞^𝑘(𝑥)_𝑘,𝑘! (34) where (𝑥)_𝑘=𝑥(𝑥−1)⋯(𝑥−𝑘+1). We have 1+𝐸_𝑞^𝑥=2^𝑥∞_𝑘=0(𝑥)_𝑘2^𝑘Δ𝑘!^𝑘𝑞,1+𝐸_𝑞^𝑥=^∞_𝑘=0𝑥𝑘²𝐹₁(𝑘−𝑥,2𝑘+1;𝑘+1;−1)+𝑥−𝑘×𝑘+1₂𝐹₁(𝑘+1−𝑥,2𝑘+2;𝑘+2;−1)×𝐸_𝑞⁻¹Δ_𝑞𝛿_𝑞^2𝑘,1+𝐸_𝑞^𝑥=1+^∞_𝑘=0𝑥×𝑘+1₂𝐹₁(𝑘+1−𝑥,2𝑘+2;𝑘+2;−1)𝐸_𝑞−𝑥−𝑘−1×𝑘+2₂𝐹₁(𝑘+2−𝑥,2𝑘+2;𝑘+3;−1)𝛿_𝑞^2𝑘. (35)(36)(37)

Finally, we present an extension of (14) using Bell polynomials (see, e.g., p. 134 in [2]) as follows: 1+𝐸_𝑞^𝑥=2^𝑥∞_𝑘=0𝑃_𝑘^(𝑥)12,12𝐿,…^𝑘𝑞,𝑘! (38) where the values of potential Bell polynomials at (1/2,1/2,…) are defined by 2^𝑥𝑃_𝑘^(𝑥)12,12,…=^∞_ℓ=0ℓ^𝑘(𝑥)_ℓ.ℓ! (39)

For a given function 𝑓(𝑡), define 𝐹_𝑞(𝑥)=∑_𝑘≥0𝑓(𝑞^𝑘)(𝑥)_𝑘/𝑘!. From (35)–(38), we obtain series transformation formulas by simply applying (35)–(37) to 𝑓: 𝐹_𝑞(𝑥)=2^𝑥∞_𝑘=0(𝑥)_𝑘2^𝑘Δ𝑘!^𝑘𝑞𝑓(1),𝐹_𝑞(𝑥)=^∞_𝑘=0𝑥𝑘²𝐹₁(𝑘−𝑥,2𝑘+1;𝑘+1;−1)+𝑥−𝑘×𝑘+1₂𝐹₁(𝑘+1−𝑥,2𝑘+2;𝑘+2;−1)×𝐸_𝑞⁻¹Δ_𝑞𝛿_𝑞^2𝑘𝑓(1),𝐹_𝑞(𝑥)=1+^∞_𝑘=0𝑥×𝑘+1₂𝐹₁(𝑘+1−𝑥,2𝑘+2;𝑘+2;−1)𝐸_𝑞−𝑥−𝑘−1𝑘+2²𝐹₁(𝑘+2−𝑥,2𝑘+2;𝑘+3;−1)×𝛿_𝑞^2𝑘𝑓(1),𝐹_𝑞(𝑥)=2^𝑥∞_𝑘=0𝑃_𝑘^(𝑥)12,12𝐿,…,^𝑘𝑞𝑓(1).𝑘! (40)(41)(42)(43)

As an example, substituting 𝑓(𝑡)=𝑡^𝑛 into (43) and noting 𝐿_𝑞𝑓(1)=^∞_𝑗=1(−1)^𝑗+1𝑗Δ^𝑗_𝑞𝑓(1)=^∞_𝑗=1(−1)^𝑗+1𝑗𝑞^𝑛−1^𝑗=log1+𝑞^𝑛−1=𝑛log𝑞, (44) yields the series transformation formula from ^∞_𝑘=0𝑞^𝑘𝑛(𝑥)_𝑘𝑘!=2^𝑥∞_𝑗=0𝑃_𝑗^(𝑥)12,12,…𝑛log𝑞^𝑗.𝑗! (45)

5. Selected Proofs

Here we present the proofs of (14)–(17) in the sense of symbolic calculus, namely, every series expansion is considered as a formal series.

For proving (14), it suffices to make use of 𝐸_𝑞=𝑒^𝐿𝑞 and (12). Indeed we have 1−𝑥𝐸_𝑞⁻¹=1−𝑥𝑒^𝐿𝑞−1=^∞_𝑘=0𝑥^𝑘𝑒^𝑘𝐿𝑞=^∞_𝑘=0𝑥^𝑘∞_𝑗=0𝑘𝐿_𝑞^𝑗=𝑗!_{∞𝑗=0∞𝑘=0}𝑥^𝑘𝑘^𝑗𝐿^𝑗_𝑞=𝑗!_∞𝑗=0𝛼_𝑗(𝑥)𝐿^𝑗_𝑞.𝑗! (46)

Equation (15) may be derived as follows: 1−𝑥𝐸_𝑞⁻¹=1−𝑥1+Δ_𝑞⁻¹=(1−𝑥)⁻¹1−𝑥Δ_𝑞1−𝑥⁻¹=^∞_𝑘=0𝑥^𝑘Δ^𝑘_𝑞(1−𝑥)^𝑘+1. (47)

To prove (16) and (17), we first establish the following lemma.

Lemma 3. Let 𝛽=1+𝛼 with 0<𝛼<1, and let 𝑥 be any real number. One has symbolic identities involving the first Gauss series as follows: 𝛽^𝑥=^∞_𝑘=0𝑥+𝑘𝛼2𝑘𝛽^2𝑘𝑘+𝑥+𝑘𝛼2𝑘+1𝛽^{2𝑘+1𝑘+1}, (48) and a modified 𝑞-form of Gauss’s first symbolic expression (cf. Section  127 of [6]): 𝐸^𝑥_𝑞=^∞_𝑘=0𝑥+𝑘Δ2𝑘^𝑞2𝑘𝐸^𝑘_𝑞+𝑥+𝑘Δ2𝑘+1^𝑞2𝑘+1𝐸_𝑞^𝑘+1=^∞_𝑘=0𝛿𝑥+𝑘2𝑘^𝑞2𝑘+Δ𝑥+𝑘2𝑘+1^𝑞𝛿_𝑞^2𝑘𝐸_𝑞−1. (49)

Proof. Starting from the following Newton’s formula: 𝛽^𝑥=^∞_𝑘=0𝑥𝑘𝛼^𝑘, (50) we multiply (𝛼+1)/𝛽=1 to the summation from the term 𝛼 up and obtain 𝛽^𝑥𝑥1𝛼=1+𝛽+𝑥1𝛼²𝛽+^∞_𝑘=2𝑥𝑘𝛼^𝑘(1+𝛼)𝛽𝑥1𝛼=1+𝛽+2𝑥+1𝛼²𝛽+^∞_𝑘=2𝛼^𝑘+1𝛽𝑥𝑘+𝑥𝑥1𝛼𝑘+1=1+𝛽+2𝑥+1𝛼²𝛽+^∞_𝑘=2𝑥+1𝛼𝑘+1^𝑘+1𝛽𝑥1𝛼=1+𝛽+2𝑥+1𝛼²𝛽+3𝑥+1𝛼³𝛽+^∞_𝑘=3𝑥+1𝛼𝑘+1^𝑘+1𝛽. (51) Repeating the operation on the series from the term 𝛼³ up yields 𝛽^𝑥𝑥1𝛼=1+𝛽+2𝑥+1𝛼²𝛽+3𝑥+1𝛼³𝛽²+3𝑥+1𝛼⁴𝛽²+^∞_𝑘=3𝑥+1𝛼𝑘+1^𝑘+1(1+𝛼)𝛽²𝑥1𝛼=1+𝛽+2𝑥+1𝛼²𝛽+3𝑥+1𝛼³𝛽²+4𝑥+2𝛼⁴𝛽²+^∞_𝑘=3𝑥+2𝛼𝑘+2^𝑘+2𝛽². (52) The above operation is repeated from 𝛼⁵ up, and so on. We obtain 𝛽^𝑥=^∞_𝑘=0𝑥+𝑘𝛼2𝑘𝛽^2𝑘𝑘+𝑥+𝑘𝛼2𝑘+1𝛽^{2𝑘+1𝑘+1}. (53) Substituting 𝛽=𝐸_𝑞 and 𝛼=Δ_𝑞 into the above identity, we obtain the desired result.

Equations (16) and (17) can be proved using the first Gauss symbolic expression (49) and the following 𝑞-form of the Everett’s symbolic expression (cf. [6], Section  129), respectively. 𝐸^𝑥_𝑞=^∞_𝑘=0𝑥+𝑘Δ2𝑘+1^𝑞2𝑘𝐸_𝑞^𝑘−1−𝑥+𝑘−1Δ2𝑘+1^𝑞2𝑘𝐸^𝑘_𝑞=^∞_𝑘=0𝐸𝑥+𝑘2𝑘+1^𝑞𝛿_𝑞^2𝑘−𝛿𝑥+𝑘−12𝑘+1^𝑞2𝑘. (54) Indeed, using (49) and noting the identity ^∞_𝑚=𝑘𝑚𝑘𝑥^𝑚=𝑥^𝑘(1−𝑥)^𝑘+1(|𝑥|<1), (55) one may derive (16) as follows: 1−𝑥𝐸_𝑞⁻¹=^∞_𝑗=0𝑥𝐸_𝑞^𝑗=^∞_𝑘=0^∞_𝑗=0𝑥𝑗+𝑘2𝑘^𝑗Δ_𝑞^2𝑘𝐸^𝑘_𝑞+^∞_𝑗=0𝑥𝑗+𝑘2𝑘+1^𝑗Δ_𝑞^2𝑘+1𝐸_𝑞^𝑘+1=^∞_𝑘=0𝑥^𝑘(1−𝑥)^2𝑘+1Δ_𝑞^2𝑘𝐸^𝑘_𝑞+𝑥^𝑘+1(1−𝑥)^2𝑘+2Δ_𝑞^2𝑘+1𝐸_𝑞^𝑘+1=^∞_𝑘=0𝑥(1−𝑥)^2𝑘+11−𝑥𝑥Δ_𝑞^2𝑘𝐸^𝑘_𝑞+Δ_𝑞^2𝑘+1𝐸_𝑞^𝑘+1=^∞_𝑘=0𝑥(1−𝑥)^2𝑘+1𝑥⁻¹Δ_𝑞^2𝑘𝐸^𝑘_𝑞−Δ_𝑞^2𝑘𝐸_𝑞^𝑘+1. (56) Equation (17) can be proved similarly using (54). However, it can also be verified by a direct symbolic computations. In fact, we have RHSof()𝑥17=1+(1−𝑥)^2∞_𝑘=0𝑥(1−𝑥)^2𝑘Δ²_𝑞𝐸_𝑞^𝑘𝐸_𝑞𝑥−𝑥=1+(1−𝑥)²𝐸_𝑞−𝑥1−𝑥/(1−𝑥)²Δ²_𝑞/𝐸_𝑞𝑥𝐸=1+^𝑞−𝑥(1−𝑥)²−𝑥Δ²_𝑞/𝐸_𝑞𝐸=1+^𝑞𝑥𝐸_𝑞−𝑥(1−𝑥)²𝐸_𝑞−𝑥𝐸_𝑞−1²𝐸=1+^𝑞𝑥1−𝑥𝐸_𝑞=1−𝑥𝐸_𝑞⁻¹=LHSof(17). (57) This completes the proofs of (14)–(17).

The proof of (35) is straightforward: 1+𝐸_𝑞^𝑥=2+Δ_𝑞^𝑥=2^𝑥Δ1+^𝑞2^𝑥=2^𝑥_𝑘≥0(𝑥)_𝑘𝑘!2^𝑘Δ^𝑘_𝑞. (58)

To prove (36), we use (49) as follows: 1+𝐸_𝑞^𝑥=1+_𝑗≥1𝑥𝑗𝐸^𝑗_𝑞=1+_𝑗≥1𝑥𝑗^𝑘≥0𝑘+𝑗Δ2𝑘^𝑞2𝑘𝐸^𝑘_𝑞+𝑘+𝑗Δ2𝑘+1^𝑞2𝑘+1𝐸_𝑞^𝑘+1=_𝑘≥0Δ_𝑞^2𝑘𝐸^𝑘_{𝑞𝑗≥𝑘}𝑥𝑗+𝑘+𝑗Δ2𝑘^𝑞2𝑘+1𝐸_𝑞^𝑘+1_{𝑗≥𝑘+1}𝑥𝑗=𝑘+𝑗2𝑘+1_𝑘≥0𝑥𝑘²𝐹₁()Δ𝑘−𝑥,2𝑘+1;𝑘+1;−1^𝑞2𝑘𝐸^𝑘_𝑞+𝑥𝑘+1²𝐹₁()Δ𝑘+1−𝑥,2𝑘+2;𝑘+2;−1^𝑞2𝑘𝐸_𝑞^𝑘+1, (59) which implies (36). Equation (37) can be proved similarly using Everett’s symbolic expression (54).

For (38), we first have 1+𝐸_𝑞^𝑥=1+𝑒^𝐿𝑞𝑥=_𝑗≥0𝑥𝑗𝑒^𝑗𝐿𝑞=_𝑗≥0𝑥𝑗^𝑘≥0𝑗𝐿_𝑞^𝑘𝑘!=_{𝑘≥0𝑗≥0}(𝑥)_𝑗𝑗^𝑘𝐿𝑗!^𝑘𝑞.𝑘! (60)(61) Using (39), we may write the part in the parenthesis of the rightmost term as 2^𝑥𝑃_𝑘^(𝑥)(1/2,1/2,…) to finish.

Acknowledgments

The authors would like to express their gratitude to the editor and anonymous referees for their very helpful comments and suggestions.