Contiguous relations, basic hypergeometric functions, and orthogonal polynomials. II. Associated big q-Jacobi polynomials

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

doi:10.1016/0022-247x(92)90360-p

Cited by 22 publications

(26 citation statements)

References 10 publications

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“…For the interested reader on this result and various other continued fraction expansions, [11,13,14,15,27] and references therein may be useful. In particular, the above continued fraction can be obtained as a limiting case of a continued fraction available in [11, p. 488 …”

Section: á Baricz and A Swaminathanmentioning

confidence: 99%

Mapping properties of basic hypergeometric functions

Baricz¹,

Swaminathan²

2014

Journal of Classical Analysis

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Section: á Baricz and A Swaminathanmentioning

confidence: 99%

Mapping properties of basic hypergeometric functions

Baricz¹,

Swaminathan²

2014

Journal of Classical Analysis

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“…Method of minimal solutions. This method, extensively used by Masson [31][32][33] and his collaborators [17][18], [21], [26], relies on the following ideas. A solution X (s) n of the 3-term recurrence relation Pincherle's theorem (see [31] and the references therein) states that (2.20) is a necessary and sufficient condition for the convergence of the continued fraction that corresponds to (2.19), namely,…”

Section: However If We Had Replacedmentioning

confidence: 99%

“…When a minimal solution exists it is unique up to a multiple independent of n. However, it was noted in [33] by means of an example that the minimal solution may change in different parts of the complex plane or from n ≥ 0 to n ≤ 0. An example of how to construct a minimal solution, when it exists, by first deriving a set of solutions of (2.19) (any two of them being linearly independent) is given in [17].…”

Section: However If We Had Replacedmentioning

confidence: 99%

The Associated Classical Orthogonal Polynomials

Rahman

2001

Special Functions 2000: Current Perspective and Future Directions

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The associated orthogonal polynomials {p n (x; c)} are defined by the 3-term recurrence relation with coefficients A n , B n , C n for {p n (x)} with c = 0, replaced by A n+c , B n+c and C n+c , c being the association parameter. Starting with examples where such polynomials occur in a natural way some of the well-known theories of how to determine their measures of orthogonality are discussed. The highest level of the family of classical orthogonal polynomials, namely, the associated Askey-Wilson polynomials which were studied at length by Ismail and Rahman in 1991 is reviewed with special reference to various connected results that exist in the literature.

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“…Recurrence relation (3.1) can be identified with the recurrence relation of the associated little q-Jacobi polynomials, [5]. The latter work gets the little q-Jacobi polynomials as limiting cases of the associated big q-Jacobi polynomials and does not give an explicit representation for the polynomials.…”

Section: The Polynomial Solution Of the Recurrence Equation For Thementioning

confidence: 99%

The spectrum of an integral operator in weighted L₂spaces

Ismail¹,

Simeonov²

2001

Pacific J. Math.

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We find the spectrum of the inverse operator of the qdifference operator D q,x f (x) = (f (x) − f (qx))/(x(1 − q)) on a family of weighted L 2 spaces. We show that the spectrum is discrete and the eigenvalues are the reciprocals of the zeros of an entire function. We also derive an expansion of the eigenfunctions of the q-difference operator and its inverse in terms of big q-Jacobi polynomials. This provides a q-analogue of the expansion of a plane wave in Jacobi polynomials. The coefficients are related to little q-Jacobi polynomials, which are described and proved to be orthogonal on the spectrum of the inverse operator. Explicit representations for the little q-Jacobi polynomials are given.

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Contiguous relations, basic hypergeometric functions, and orthogonal polynomials. II. Associated big q-Jacobi polynomials

Cited by 22 publications

References 10 publications

Mapping properties of basic hypergeometric functions

Mapping properties of basic hypergeometric functions

The Associated Classical Orthogonal Polynomials

The spectrum of an integral operator in weighted L₂spaces

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Contiguous relations, basic hypergeometric functions, and orthogonal polynomials. II. Associated big q-Jacobi polynomials

Cited by 22 publications

References 10 publications

Mapping properties of basic hypergeometric functions

Mapping properties of basic hypergeometric functions

The Associated Classical Orthogonal Polynomials

The spectrum of an integral operator in weighted L2spaces

Contact Info

Product

Resources

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The spectrum of an integral operator in weighted L₂spaces