The Scientific World Journal

Volume 2014 (2014), Article ID 940358, 13 pages

http://dx.doi.org/10.1155/2014/940358

Research Article

New Proofs of Some -Summation and -Transformation Formulas

Xian-Fang Liu, Ya-Qing Bi, and Qiu-Ming Luo

Department of Mathematics, Chongqing Higher Education Mega Center, Chongqing Normal University, Huxi Campus, Chongqing 401331, China

Received 13 February 2014; Accepted 11 April 2014; Published 7 May 2014

Academic Editor: Kishin Sadarangani

Copyright © 2014 Xian-Fang Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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

1. Introduction

The probabilistic method is an important tool to derive results in combinatorics, theory of numbers, and other fields (see [1–15]). There have been many applications in the basic hypergeometric series (or -series). For example, Fulman [3] presented a probabilistic proof of Rogers-Ramanujan identity using Markov chain. Chapman [2] proved the Andrews-Gordon identity by using extended Fulman's methods. Kadell [4] gave a probabilistic proof of Ramanujan's summation based on the order statistics.

Recently, Wang [13, 14] constructed a random variable and introduced a new probability distribution :

where

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 , 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 .

The -shifted factorials are defined by

Clearly,

The following are compact notations for the multiple -shifted factorials:

The basic hypergeometric series or -series are defined by (see [16, 17])

Heine introduced the 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 :

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.

Lemma 1 (see [21, page 5. (2.5)] [20, Theorem 1]). One has

provided and , provided that there are no zero factors in the denominator of the integrals.

Lemma 2 (see [21, page 5. (2.7)]). One has

provided and , 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 of -series 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 . 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.

2. 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.

Theorem 3. Let denote a random variable with probability distribution , . Then one has

provided that , , and .

Proof. A random variable has the distribution . From definitions (9) we have

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.

Theorem 4. Let denote a random variable with probability distribution , . Then one has

provided that , , and .

Proof. By (7) and (8) we have

By (4) we have

Substituting (20) and (21) into the right-hand side of (14), we obtain

Next, let us replace by , respectively, and let in (22); we get

The proof is complete.

Theorem 5. Let denote a random variable with probability distribution , . Then one has

provided that , , and .

Proof. Letting in (14) of Theorem 3, we obtain (24).

Corollary 6 (see [13, page 463, Theorem 1]). Let denote a random variable with probability distribution , . Then one has

provided that .

Proof. Using (24) of Theorem 5, we deduce

Using (31) of Theorem 8, we have

Substituting (27) into the right-hand sides of (26), we have

The proof is complete.

Corollary 7 (see [14, page 245, Lemma 2.4]). Let denote a random variable with probability distribution , . Then one has

provided that .

Proof. Letting or in (14) of Theorem 3, then we have

The proof is complete.

3. Probabilistic Proofs of Transformation Formulas of

Sears’ transformation formula is widely applied to the special functions. In this section we will introduce probabilistic proofs of transformation of .

Theorem 8 (see [17, page 359. III. 9, III. 10]). One has

Proof. Interchanging and in (14), then we have

Interchanging and in (14), then we have

By (14) and (33), we obtain

and, replacing by in (35), we obtain a transformation formula

By (14) and (34) and then replacing by , we obtain (31).

By (33) and (34), we have

and, replacing by in (37), we obtain (32). The proof is complete.

4. Probabilistic Proof of Heine and Jackson’s Transformations

Heine [22] derived transformation formulas for 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.

Theorem 9 (see [17, page 359, III. 1, III. 2, III. 3]). Heine’s transformation formulas for are

Proof. Comparing (24) of Theorem 5 and (25) of Corollary 6, we obtain

or, equivalently, that

Setting in (42), we have

Replacing by in (43), we get

which is just (38).

Setting and and replacing by in (14), we have

Setting and in (14) of Theorem 3, we have

Comparing (45) and (46), we obtain

Replacing by , we get

Letting in (48) gives

We get (39). From (39) we can deduce (40).

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 and .

Theorem 10 (see [17, page 359, III.4]). Jackson’s transformations of , series are

Proof. This includes employing two different forms of .

Letting in (14) of Theorem 3 and then replacing by , we get

Comparing (51) and (19) of Theorem 4 gives

Then we obtain

Replacing by gives

This completes the proof.

5. 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. The -binomial 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 a -analogue 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.

Theorem 11 (see [16, page 488, Theorem ] [17, page 354. II. 3]). The -binomial theorem is

Proof. Below we give two proofs of (55).

Setting and replacing and by and in (14), we obtain

Comparing (56) and (29) of Corollary 7, we have

Then we obtain

Replacing by , we can get

that is,

Another proof of the -binomial theorem is as follows.

Setting and and replacing by in (14), we obtain

Letting or and in (14) of Theorem 3, we obtain

Comparing (61) and (62) gives

Then we obtain

Replacing by gives

that is,

This proof is complete.

Theorem 12 (see [17, page 354, II. 7]). The -Chu-Vandermonde sums are

Proof. The below are two proofs of the -Chu-Vandermonde.

(i)First proof: setting and replacing by in (14), we have

Replacing by in (68), then we have

where

Hence,

By using the probability distribution and employing Andrews-Askey -integral (11), now we calculate the expectation of the random variables as follows:

Comparing (71) and (72) gives

Then we obtain

which is just -Vandermonde sums (67).

(ii)Second proof: replacing by in (29), we have

Comparing (71) and (75), we obtain

Hence,

which is just -Vandermonde sums (67).

Theorem 13 (see [16, page 522, Corollary ] or [17, page 354, II. 8]). The -Gauss sum is

Proof. Letting and replacing by in (14), we obtain

Comparing (29) and (79) gives

hence we get

Replacing by in the above formula, we obtain

which is just the -Gauss sum (78).

Theorem 14 (see [17, page 354, II. 9]). The -Kummer sum formula is

Proof. Letting in (14) and then replacing by , we have

Replacing by in (84), we write

where

Hence, we obtain

By using the probability distribution and Lemma 2, we calculate the expectation of the random variables as follows:

Comparing (87) and (88), we have

Using Heine’s transformation and -binomial theorem, we have

Hence, we obtain (83).

Theorem 15 (see [17, page 354, II. 10]). Bailey’s sum formula is

Proof. By (19), we have

Replacing by in (92) gives

where

Hence, we have

By using the probability distribution and Lemma 2, we calculate the expectation of the random variables as follows:

Comparing (95) and (96), we have

Hence, we get (91).

Theorem 16 (see [17, page 354, II. 11]). The Gauss sum formula is

Proof. By (14), we have

Replacing by in (99) gives

where

Hence,

By using the probability distribution and employing Andrews-Askey -integral (13) of Lemma 2, we calculate the expectation of the random variables as follows:

Comparing (102) and (103), we have

Using Heine’s transformation formula, we have

Substituting (105) into (104) yields

Noting that

we have

Substituting (108) into (106) yields

Hence, we obtain (98).

Theorem 17 (see [17, page 354, II. 5]). A sum formula of is

Proof. Letting in (19) of Theorem 4, we obtain

Letting and in (14) of Theorem 3 or setting and and replacing by in (14) of Theorem 3, we obtain

Comparing (111) and (112) gives

that is,

Replacing by , we get

This proof is complete.

Theorem 18 (see [17, page 354, II. 1, II. 2]). The two -exponential functions are

Proof. Setting and then replacing by in (14), we obtain

Letting and in (14) or letting and and replacing by in (14), we obtain

Comparing (117) and (118) gives

From the above formula, we obtain

Replacing by gives

that is,

Similarly, setting and in (19) of Theorem 4, we obtain

And obviously letting and in (14) of Theorem 3, we obtain

Comparing (123) and (124), we obtain

Replacing by , we get

that is,

Remark 19. In the present paper we obtain the part transformation and sum formulas of the -series by applying the probabilistic method. We hope to find and construct another probability distribution in order to prove the transformation and sum formulas of the bilateral basic hypergeometric series, for example, Ramanujan and Bailey sum formulas and so forth.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors express their sincere gratitude to the referee for many valuable comments and suggestions. The present investigation was supported by the Natural Science Foundation Project of Chongqing, China, under Grant CSTC2011JJA00024, Research Project of Science and Technology of Chongqing Education Commission, China, under Grant KJ120625, and Fund of Chongqing Normal University, China, under Grants 10XLR017 and 2011XLZ07.

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