New Proofs of Some q-Summation and q-Transformation Formulas

We obtain an expectation formula and give the probabilistic proofs of some summation and transformation formulas of q-series based on our expectation formula. Although these formulas in themselves are not the probability results, the proofs given are based on probabilistic concepts.

By applying the above probability distribution, Wang proved the -binomial theorem and -Gauss summation formula and also obtained some new summation formulas and transformation formulas. One of the most important concepts in probability theory is that of the expectation of a random variable. If is a discrete random variable having a probability mass function ( ), then the expectation, or the expected value, or the expectation operator of , denoted by E[ ], is defined by (e.g., [9, page 125]) In the following section we introduce some notations, definitions, and formulas of -series. Throughout this paper we suppose ∈ C, | | < 1.
The -shifted factorials are defined by 2 The Scientific World Journal The following are compact notations for the multipleshifted factorials: ( 1 , 2 , . . . , ; ) = ( 1 ; ) ( 2 ; ) ⋅ ⋅ ⋅ ( ; ) , The basic hypergeometric series or -series are defined by (see [16,17]) Heine introduced the +1 basic hypergeometric series which is defined by Jackson defined the -integral (see [17,18]): The following is the Andrews-Askey integral (see [19]) which can be derived from Ramanujan's 1 1 : provided that there are no zero factors in the denominator of the integrals. Recently, Liu and Luo [20] further generalized the above Andrews-Askey integral in the following more general form.
provided | ℎ| < 1 and | | < 1, provided that there are no zero factors in the denominator of the integrals.
provided | ℎ| < 1 and | | < 1, provided that there are no zero factors in the denominator of the integrals.
The aim of the present paper is to give an expectation formula and introduce some probabilistic proofs of the corresponding summation and transformation formulas ofseries based on an expectation formula. In Section 2 we give an expectation formula of the random variables ( ; ) ∞ / ( , , ; ) ∞ . In Section 3 we show the probabilistic proofs of transformation formulas of 3 2 . In Section 4 we give probabilistic proof of Heine's transformations and Jackson's transformations. In Section 5 we give probabilistic proof of some formulas of -series, for example, -binomial theorem, -Chu-Vandermonde sum formulas, -Gauss sum formula, -Kummer sum formula, Bailey sum formula, and so forth.

Main Theorem
In this section we obtain the expectation formulas of some random variables which are very useful to prove the summation and transformation formulas of -series.

Proof.
A random variable has the distribution ( ; ). From definitions (9) we have The Scientific World Journal 3 From definitions (10) and combining (15) we have By using the probability distribution ( ; ) and noting (16) and (12) of Lemma 1, we calculate the expectation of the random variable ( ; ) ∞ /( , , ; ) ∞ as follows: .
Hence, we obtain The proof is complete.
The proof is complete.
Proof. Using (24) Substituting (27) into the right-hand sides of (26), we have The proof is complete.
Proof. Letting = or = = in (14) of Theorem 3, then we have The proof is complete.

Probabilistic Proofs of Transformation
Formulas of 3 2 Sears' 3 2 transformation formula is widely applied to the special functions. In this section we will introduce probabilistic proofs of transformation of 3 2 .

Probabilistic Proof of Heine and Jackson's Transformations
Heine [22] derived transformation formulas for 2 1 and also proved Euler's transformation formula. A basic hypergeometric representation for a given function is by no means unique. There are groups of transformation between various hypergeometric representations of the same function. We will first prove the classical Heine's transformation formula which will be useful in proving many other formulas. In this section we give the probabilistic proofs of Heine and Jackson's transformations.
Jackson's transformations formula is an important formula in basic hypergeometric series, and now we give a probabilistic proof of Jackson's transformation formulas for 2 1 and 2 2 .    This completes the proof.

Probabilistic Proofs of Some Formulas of -Series
The -binomial theorem is an important mathematical result which has been widely applied in the special functions, physics, quantum algebra, and quantum statistics. Thebinomial theorem was derived by Cauchy [23], Heine [22], and Jacobi [24] concerning the nonterminating form. There are many proofs of the -binomial theorem to show the corresponding references; for example, a better and simpler proof, by using the method of the finite difference, was obtained by Askey (see [25]); a nice proof of the -binomial theorem based on combinatorial considerations was given by Joichi and Stanton (see [26]). In 1847, Heine [22] derived aanalogue of Gauss's summation formula which is important in -series. Joichi and Stanton [26] gave a bijective proof of the -Gauss summation formula based on combinatorial considerations. Rahman and Suslov [27] used the method of the first order linear difference equations to prove the -Gauss summation formula. By analytic continuation, the terminating case, when = − , reduces to -analogues of Vandermonde's formula. Bailey and Daum independently discovered the -Kummer summation formula.
In this section we will introduce probabilistic proof of some formulas of -series, for example, -binomial theorem, -Chu-Vandermonde, -Gauss summation formula and -Kummer summation formula, and so forth.
Proof. Below we give two proofs of (55). Setting = = 0 and replacing and by and in (14), we obtain Replacing by , we can get Another proof of the -binomial theorem is as follows.
The Scientific World Journal 7 Setting = = 0 and = and replacing by in (14), we obtain Comparing (61) and (62) gives Then we obtain This proof is complete.
Proof. Letting = = and replacing by in (14), we obtain which is just the -Gauss sum (78).
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