Abstract
In this paper, we establish one general q-exponential operator identity by solving one simple q-difference equation. Using this q-difference equation, we get some generalizations of Andrews-Askey and Askey-Wilson integral. In addition, we also discuss some properties of q-polynomials .
MSC:33D05, 33D45, 11B65, 33D60.
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1 Introduction and notations
For decades, various families of q-polynomials and q-integral have been investigated rather widely and extensively due mainly to their having been found to be potentially useful in such wide variety of fields as theory of partitions, number theory, combinatorial analysis, finite vector spaces, Lie theory, etc. (cf. [1–27]). There are many techniques to achieve the ends; for instance, analysis methods (cf. [3–6, 14, 16]), combinatorics method (cf. [17]), and q-operator method (cf. [7, 9–11, 19, 24]) and so on. In resent years, the authors [8, 20–22] derived some formulas of q-polynomials and q-integral from studying the properties of solutions about some q-difference equations. Inspired by their work, in this paper, we will present one more generalized q-difference equation and give some applications of it.
We adopt the notations used by Gasper and Rahman [15]. Throughout the paper unless otherwise stated we assume that . Let ℕ denote the set of non-negative integer, ℂ denote the set of complex numbers.
For any complex number a, the q-shifted factorial are defined as
and we also adopt the following compact notation for the multiple q-shifted factorial:
The basic hypergeometric series is given by
For any function of one variable, the q-derivative of with respect to x is defined as (cf. [7–11, 18–22])
and we further defined , and for , .
The q-binomial coefficient is defined as
For , we define the following generalized q-operator:
Some special cases of the above q-operator had been studied by many researchers. For instance, the authors [9, 19–21, 24] made a systematic study on . Some applications of were given in [10, 11, 22]. Some properties and applications of were discussed in [8, 11]. In this paper, we present the following more generalized q-difference equation for the above q-operator.
Theorem 1.1 Let be a -variable analytic function in a neighborhood of , , satisfying the q-difference equation
where
Then we have
Corollary 1.2 ([8], Eq. (1.12))
Let be a 5-variable analytic function in a neighborhood of , satisfying the q-difference equation
then
Remark 1.3 Letting , , Eq. (6) reduces to (9). Setting , , then replacing b, c by a, b respectively, Eq. (6) reduces to [20], Theorem 1. Putting , , then replacing , b, c by a, c, b, respectively, Eq. (6) reduces to [22], Proposition 1.2.
Proof of Theorem 1.1 From the theory of several complex variables in [28] (or [25], p.28, Hartog’s theorem), we assume that
and then substitute the above equation into (6) yielding
Equating the coefficients of , we have
For each , we get
By iteration, we find that
Putting in (11), we get . Substituting (15) into (11), we get (8). This completes the proof. □
Theorem 1.4 If , , , , then
Proof We use to denote the right side of (16). We have
and , () are defined as (7), we have
Replacing by n, then applying (4), we find that the above equation is equal to
So satisfies (6), applying (8), we complete the proof. □
Letting , , , in Eq. (III.12) ([15], p.360), we have
Combining the above identity and (16), we find the following generalized formula of [11], Lemma 2.3 (or [8], Eq. (3.4)).
Corollary 1.5 If , , , , then
Letting in (16), we find the following.
Corollary 1.6 If , , , , then
Letting in (16), we find the following.
Corollary 1.7 If , , , then
Letting in (16), then applying q-Chu-Vandermonde summation ([15], p.354, Eq. (II.6))
and we obtain the following.
Corollary 1.8 If , , , then
Remark 1.9 It were difficult to distinguish analysis of the functions of the right side of (16) (or (21), (22)) if we would remove the condition . But in (23) and (25), we do not need the condition . Under , it is easy to verify that the right sides of (23) and (25) are analytic functions in a neighborhood of . In this paper, the symbols and are frequently used. Here is defined as (17), and is equal to .
The paper is organized in the following manner. In the next two sections we give some generalizations of Andrews-Askey and Askey-Wilson integrals by the q-difference equation. In Section 4, we discuss some properties of q-polynomials . Several special cases and examples of our results are also pointed out, in the concluding section.
2 Generalizations of Andrews-Askey integrals
We have
provided that there are no zero factors in the denominator of the integral, which could be directly derived from Andrews-Askey integrals ([3], Eq. (2.1) or [5], Eq. (1.15)) after some simple replacing. In [8] (or [21, 26, 27]), some generalizations and applications of (26) are given. In this paper we give the following generalizations of the above identity.
Theorem 2.1 If , , , , then
Applying (20), we rewrite (27) as follows.
Corollary 2.2 If , , , , then
If , , then letting , the left-hand side of (28) is equal to
For , the inner summation is equal to
Substituting the above identity into (29), we have the following.
Corollary 2.3 ([8], Theorem 14)
If , , , then
Remark 2.4 For , we find that
So we see that the identity (31) is the same as Theorem 14 in [8] after replacing by , respectively.
Proof of Theorem 2.1 We rewrite (27) as follows:
If we use we have
In the same way as proving (16), we can verify satisfies (6), so we have
We use and we have
It is easy to prove satisfies (6), so we find that
Combining the above identity and (26), we complete the proof of (27). □
Theorem 2.5 If , , , , then
Proof We rewrite (38) as follows:
Setting and denoting the left-hand and the right-hand side of (39), respectively, and taking in proving of Theorem 1.4, we can verify both and satisfy (6). Letting , from (25), we get
This completes the proof. □
Theorem 2.6 If , , , , then
Proof Letting
and
we can easily verify both of the above identities satisfy (6), so we have
and
Combining (26), we complete the proof of (41). □
Interchanging a and b in (41), we get
Combing the above identity and (41), then replacing by , respectively, we recover the special case for in Eq. (III.11) ([15], p.360).
Corollary 2.7 ([15], p.360, Eq. (III.11))
We have
Theorem 2.8 If , , , , , , then
Proof Letting
and
we can easily verify both of the above identities satisfy (6), so we have
and
From (41), we conclude that
By (51) and (52), we complete the proof. □
Interchanging a and b in (48), similar to (47), we find the following.
Corollary 2.9 We have
Setting , in (54), then letting , we have the following.
Corollary 2.10 If , then
where , .
3 Generalizations of Askey-Wilson integral
In [12], we had derived a new of q-contour integral formula from the following elegant Askey-Wilson integral formula (cf. [6], Theorem 2.1):
where the contour C is a deformation of unit circle so that the poles of lie outside the contour and the origin and poles of lie inside the contour. In this section, we get the following generalizations of the above equation.
Theorem 3.1 If , , , , then
Proof We rewrite (57) as follows:
We use and to denote the left-hand and the right-hand side of (58), respectively. By the same method as in Theorem 1.4, we can verify they both satisfy (6). Letting , we have
Applying (16), the above identity is equal to the left side of (58). This completes the proof. □
Employing the above theorem, using q-operator , similar to the above proof, we conclude the following.
Theorem 3.2 If , , , , , , then
4 Some properties of q-polynomials
For , , we define
We can get some famous polynomials from , e.g., letting , , , the polynomials reduce to the classical Al-Salam-Carlitz polynomials (cf. [1], Eq. (1.11)),
Setting , in (61), then letting , we have
Taking , , , , , , , in (63), we have the Askey-Wilson polynomials ([6], Eq. (1.15))
Putting , , , , , , , in (63), we get the q-Racah polynomials ([15], Eq. (7.2.17)),
In this section, we will give some properties of q-polynomials by q-difference equation. We now show the satisfies the following q-difference equation.
Theorem 4.1 If , , then
where , are defined as (7).
Proof Letting defined as (17), and denoting , we have
Replacing by k, we find that the above equation is equal to
On the other hand
This completes the proof. □
For satisfies (6), applying (8), we find the following.
Corollary 4.2 If is defined as (61), then
Combining the above equation and (23), we obtain the following generating functions for .
Theorem 4.3 If , then
Setting , , in (71), we conclude the following.
Corollary 4.4 ([1], Eq. (1.13))
If , then
Theorem 4.5 If , , , , then
To prove the above theorem, we need the following lemma.
Lemma 4.6 If , , , , then
Proof Letting denoting the right-hand side of (74), similar to the proof of Theorem 1.4, we see that the functions satisfies (6). Applying (8), we complete the proof. □
Proof of Theorem 4.5 The left-hand side of (73) is equal to
Using Lemma 4.6, we complete the proof. □
Theorem 4.7 If , , , , , then
Proof The left-hand side of (76) is equal to
By Theorem 1.4, the proof is complete. □
Letting in (76), we have the following.
Corollary 4.8 If , , , , , then
Setting in (76), then applying (24), we find the following.
Corollary 4.9 If , then
5 Some special cases
In this section, we briefly consider some consequences and special cases of the results derived in Section 2. If we take , , in (27), applying (24), we obtain the following.
Proposition 5.1 If , , , then
If we take , , , in (38), applying the q-Gauss summation ([15], p.354, Eq. (II.8))
then replacing by , respectively, we get
For series, using Hall’s transformation ([15], p.359, Eq. (III.10))
we find the following.
Proposition 5.2 ([20], Theorem 9)
We have
Setting in the above identity, we obtain the following.
Proposition 5.3 ([20], Theorem 8)
We have
Noting
then letting , , , in (41), we get the following.
Proposition 5.4 If , we get
Taking , , , in (87), then letting yields the following.
Corollary 5.5 If , then
If let , , , in (87), and setting , we have the following.
Corollary 5.6 If , then
Combining with the above two identities, we obtain the following.
Corollary 5.7 If , then
Taking , , , in (87), then letting yields the following.
Corollary 5.8 If , then
Setting , , , in (87), then letting yields the following.
Corollary 5.9 If , then
Combining with the above two identities, we obtain the following.
Corollary 5.10 If , then
Remark 5.11 The symbol denotes the largest integer ≤x.
References
Al-Salam WA, Carlitz L: Some orthogonal q -polynomials. Math. Nachr. 1965, 30: 47-61. 10.1002/mana.19650300105
Andrews GE Conference Series in Mathematics 66. In q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. Am. Math. Soc., Providence; 1986.
Andrews GE, Askey R: Another q -extension of the beta function. Proc. Am. Math. Soc. 1981, 81: 97-100.
Askey R: The q -gamma and q -beta function. Appl. Anal. 1978, 8: 125-141. 10.1080/00036817808839221
Askey R: A q -extension of Cauchy’s form of the beta integral. Q. J. Math. 1981, 32(3):255-266. 10.1093/qmath/32.3.255
Askey R, Wilson J: Some basic hypergeometric polynomials that generalize Jacobi polynomials. Mem. Am. Math. Soc. 1985., 54: Article ID 319
Bowman D: q -Differential operators, orthogonal polynomials, and symmetric expansions. Mem. Am. Math. Soc. 2002., 159: Article ID 757
Cao J: A note on generalized q -difference equations for q -beta and Andrews-Askey integral. J. Math. Anal. Appl. 2014, 412: 841-851. 10.1016/j.jmaa.2013.11.027
Carlitz L: Generating functions for certain q -orthogonal polynomials. Collect. Math. 1972, 23(2):91-104.
Fang J-P: q -Differential operator identities and applications. J. Math. Anal. Appl. 2007, 332: 1393-1407. 10.1016/j.jmaa.2006.10.087
Fang J-P: Some applications of q -differential. J. Korean Math. Soc. 2010, 47: 223-233. 10.4134/JKMS.2010.47.2.223
Fang J-P: Note on a q -contour integral formula. Appl. Math. Comput. 2014, 233: 292-297.
Fine NJ Mathematical Surveys and Monographs 27. In Basic Hypergeometric Series and Applications. Am. Math. Soc., Providence; 1988.
Gasper G, Rahman M: q -Extensions of Barnes’, Cauchy’s, and Euler’s beta integrals. In Topics in Mathematical Analysis. Edited by: Rassias YM. World Scientific, Singapore; 1989:294-314.
Gasper G, Rahman M: Basic Hypergeometric Series. 2nd edition. Cambridge University Press, Cambridge; 2004.
Ismail MEH, Masson DR: q -Hermite polynomials, biorthogonal rational functions, and q -beta integrals. Trans. Am. Math. Soc. 1994, 346: 63-116.
Ismail MEH, Stanton D, Viennot G: The combinatorics of q -Hermite polynomials and the Askey-Wilson integral. Eur. J. Comb. 1987, 8(4):379-392.
Jackson FH: On a q -definite integrals. Q. J. Pure Appl. Math. 1910, 41: 193-203.
Liu Z-G: Some operator identities and q -series transformation formulas. Discrete Math. 2003, 265: 119-139. 10.1016/S0012-365X(02)00626-X
Liu Z-G: Two q -difference equations and q -operator identities. J. Differ. Equ. Appl. 2010, 16: 1293-1307. 10.1080/10236190902810385
Liu Z-G: An extension of the non-terminating summation and the Askey-Wilson polynomials. J. Differ. Equ. Appl. 2011, 17: 1401-1411. 10.1080/10236190903530735
Lu D-Q: q -Difference equation and the Cauchy operator identities. J. Math. Anal. Appl. 2009, 359: 265-274. 10.1016/j.jmaa.2009.05.048
Medema JC, Álvarez-Nodarsea R, Marcellán F: On the q -polynomials: a distributional study. J. Comput. Appl. Math. 2001, 135: 157-196. 10.1016/S0377-0427(00)00584-7
Rogers LJ: On the expansion of some infinite products. Proc. Lond. Math. Soc. 1893, 24(1):337-352.
Taylor J Graduate Studies in Mathematics 46. In Several Complex Variables with Connections to Algebraic Geometry and Lie Groups. Am. Math. Soc., Providence; 2002.
Wang M-J: A remark on Andrews-Askey integral. J. Math. Anal. Appl. 2008, 341: 1487-1494. 10.1016/j.jmaa.2007.11.011
Wang M-J: A recurring q -integral formula. Appl. Math. Lett. 2010, 23: 256-260. 10.1016/j.aml.2009.10.004
Range RM: Complex analysis: a brief tour into higher dimensions. Am. Math. Mon. 2003, 110: 89-108. 10.2307/3647769
Acknowledgements
The author would like to thank the editors and the anonymous referees for their careful reading of the paper. The author is supported by National Natural Science Foundation of China (No. 11471138) and National Natural Science Foundation of China (No. 11371163). The author is also supported by Jiangsu Overseas Research and Training Program for University Prominent Young and Middle-Aged Teachers and Presidents, Universities Natural Science Foundation of Jiangsu (No. 14KJB110002) and SRF for ROCS, SEM.
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Fang, JP. Applications of a generalized q-difference equation. Adv Differ Equ 2014, 267 (2014). https://doi.org/10.1186/1687-1847-2014-267
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DOI: https://doi.org/10.1186/1687-1847-2014-267