Contiguous relations, basic hypergeometric functions, and orthogonal polynomials. III. Associated continuous dual q-Hahn polynomials

Gupta, Dharma P.; Ismail, Mourad E. H.; Masson, David R.

doi:10.1016/0377-0427(95)00264-2

Cited by 25 publications

(30 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This recurrence is found in [32] (but with their q as our q −1 ), and the minimal solution is shown to bẽ…”

Section: An Alternative Expression Ismentioning

confidence: 79%

“…. }, then the corresponding unilateral birth-and-death process has a specialization of the associated continuous dual q-Hahn polynomials as its related family of orthogonal polynomials [32]. However, we have not been able to "take limits as n → ∞" in the resulting spectral representation of the transition probabilities to obtain similar formulae forX.…”

Section: A Remark On Spectral Representationsmentioning

confidence: 98%

“…Closed-form expressions for continued fractions of this form are listed in Ramanujan's "lost" notebook (see the discussion in [5]), and evaluations for various ranges of the parameters (although not all the values we need) can be found in [5,32,33] (although in the last several parameter restrictions are omitted). …”

Section: Hitting Time Distributions For T Qmentioning

confidence: 99%

See 2 more Smart Citations

Brownian motion on disconnected sets, basic hypergeometric functions, and some continued fractions of Ramanujan

Bhamidi¹,

Evans²,

Peled³

et al. 2008

Institute of Mathematical Statistics Collections

View full text Add to dashboard Cite

Motivated by Lévy's characterization of Brownian motion on the line, we propose an analogue of Brownian motion that has as its state space an arbitrary closed subset of the line that is unbounded above and below: such a process will be a martingale, will have the identity function as its quadratic variation process, and will be "continuous" in the sense that its sample paths don't skip over points. We show that there is a unique such process, which turns out to be automatically a reversible Feller-Dynkin Markov process. We find its generator, which is a natural generalization of the operator f → 1 2 f ′′ . We then consider the special case where the state space is the self-similar set {±q k : k ∈ Z} ∪ {0} for some q > 1. Using the scaling properties of the process, we represent the Laplace transforms of various hitting times as certain continued fractions that appear in Ramanujan's "lost" notebook and evaluate these continued fractions in terms of basic hypergeometric functions (that is, q-analogues of classical hypergeometric functions). The process has 0 as a regular instantaneous point, and hence its sample paths can be decomposed into a Poisson process of excursions from 0 using the associated continuous local time. Using the reversibility of the process with respect to the natural measure on the state space, we find the entrance laws of the corresponding Itô excursion measure and the Laplace exponent of the inverse local timeboth again in terms of basic hypergeometric functions. By combining these ingredients, we obtain explicit formulae for the resolvent of the process. We also compute the moments of the process in closed form. Some of our results involve q-analogues of classical distributions such as the Poisson distribution that have appeared elsewhere in the literature.

show abstract

“…This recurrence is found in [32] (but with their q as our q −1 ), and the minimal solution is shown to bẽ…”

Section: An Alternative Expression Ismentioning

confidence: 79%

Section: A Remark On Spectral Representationsmentioning

confidence: 98%

See 1 more Smart Citation

Brownian motion on disconnected sets, basic hypergeometric functions, and some continued fractions of Ramanujan

Bhamidi¹,

Evans²,

Peled³

et al. 2008

Institute of Mathematical Statistics Collections

View full text Add to dashboard Cite

show abstract

“…Very recently the Kummer identity has been generalized by Vidúnas [12], by using the contiguous relation. For more details about hypergeometric series and their contiguous relations see [1,2,[13][14][15][16][17][18][19][20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

On some new contiguous relations for the Gauss hypergeometric function with applications

Rakha

Rathie²,

Chopra³

2011

Computers & Mathematics with Applications

View full text Add to dashboard Cite

a b s t r a c tContiguous relations for hypergeometric series contain an enormous amount of hidden information. Applications of contiguous relations range from the evaluation of hypergeometric series to the derivation of summation and transformation formulas for such series. In this paper, a new set of contiguous function relations are established. Applications of such relations to hypergeometric summation formulas and the theory of Jacobi polynomials are presented.

show abstract

“…In [9], some consequences of the contiguous relations of 2 F 1 were proved, while in [10], a new method of the shifted operators for computing the contiguous relations of 2 F 1 was introduced. More details about contiguous relations and their application can be found in [11][12][13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

Contiguous relations and their computations for F12 hypergeometric series

Ibrahim

Rakha

2008

Computers & Mathematics with Applications

View full text Add to dashboard Cite

a b s t r a c tThe hypergeometric function 2 F 1 [a 1 , a 2 ; a 3 ; z] plays an important role in mathematical analysis and its application. Gauss defined two hypergeometric functions to be contiguous if they have the same power-series variable, if two of the parameters are pairwise equal, and if the third pair differs by ±1. He showed that a hypergeometric function and any two other contiguous to it are linearly related. In this paper, we present an interesting formula as a linear relation of three shifted Gauss polynomials in the three parameters a 1 , a 2 and a 3 . More precisely, we obtained a recurrence relation including ] for any arbitrary integers α 1 , α 2 and α 3 .

show abstract

Contiguous relations, basic hypergeometric functions, and orthogonal polynomials. III. Associated continuous dual q-Hahn polynomials

Abstract: Journal of Computational and Applied Mathematics 68 (1996) 115-149. doi:10.1016/0377-0427(95)00264-22016-03-04T18:47:27

Cited by 25 publications

References 16 publications

Brownian motion on disconnected sets, basic hypergeometric functions, and some continued fractions of Ramanujan

Brownian motion on disconnected sets, basic hypergeometric functions, and some continued fractions of Ramanujan

On some new contiguous relations for the Gauss hypergeometric function with applications

Contiguous relations and their computations for F12 hypergeometric series

Contact Info

Product

Resources

About