(𝑛+1,𝑚+1)-hypergeometric functions associated to character algebras

Mizukawa, Hiroshi; Tanaka, Hajime

doi:10.1090/s0002-9939-04-07399-x

Cited by 37 publications

(41 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…An additional comment. After this paper was completed we became aware of a recent arXiv posting [14], where the authors point out some important work of H. Mizukawa and H. Tanaka [19]. In a future publication we return to the probabilistic origin of the work of M. Hoare and M. Rahman and we discuss the relation between the approach in [19], based on the notion of character algebras, and our own.…”

Section: Straightforward Algebra Givesmentioning

confidence: 96%

On a Family of 2-Variable Orthogonal Krawtchouk Polynomials

Grünbaum

Rahman

2010

SIGMA

View full text Add to dashboard Cite

Abstract. We give a hypergeometric proof involving a family of 2-variable Krawtchouk polynomials that were obtained earlier by Hoare and Rahman [SIGMA 4 (2008), 089, 18 pages] as a limit of the 9 − j symbols of quantum angular momentum theory, and shown to be eigenfunctions of the transition probability kernel corresponding to a "poker dice" type probability model. The proof in this paper derives and makes use of the necessary and sufficient conditions of orthogonality in establishing orthogonality as well as indicating their geometrical significance. We also derive a 5-term recurrence relation satisfied by these polynomials.

show abstract

Section: Straightforward Algebra Givesmentioning

confidence: 96%

On a Family of 2-Variable Orthogonal Krawtchouk Polynomials

Grünbaum

Rahman

2010

SIGMA

View full text Add to dashboard Cite

show abstract

“…a special case of Gelfand hypergeometric function [2,4], was known to, among others, the Japanese authors Mizukawa and Tanaka [15], who used them to prove their orthogonality in some special cases. Later, Mizukawa [13] gave a complete orthogonality proof of (1.9) using Gelfand pairs, then [15] used character algebras and closely following this proof came Iliev and Terwilliger's proofs, first for n = 2 [11], then for general n in [12], in which they used tools from Lie algebra theory. In these proofs the authors found it convenient to use 1−u ij instead of u ij as parameters in (1.9), and also to use…”

Section: Introductionmentioning

confidence: 99%

“…Griffiths [5] as early as 1971 which he defined as coefficients in an expansion of their generating function. However, Mizukawa and Tanaka [15] seem to have been the first to give the explicit expression in (1.9).…”

Section: Introductionmentioning

confidence: 99%

“…For a very rich and recent extension of the scalar valued solution of the hypergeometric equation, see [16]. It is worth noticing that in this case there is yet no extension of the Euler integral representation formula for the 2 F 1 function, and that algebraic methods such as those coming from Lie algebras or group representation theory, which have played such in important role in [11,12,13,14,15] have not made a mark in this matrix valued extension yet. Similar consideration would be relevant if one were to consider a matrix valued extension of, for instance, the work in [3].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A System of Multivariable Krawtchouk Polynomials and a Probabilistic Application

Grünbaum

Rahman

2011

SIGMA

View full text Add to dashboard Cite

Abstract. The one variable Krawtchouk polynomials, a special case of the 2 F 1 function did appear in the spectral representation of the transition kernel for a Markov chain studied a long time ago by M. Hoare and M. Rahman. A multivariable extension of this Markov chain was considered in a later paper by these authors where a certain two variable extension of the F 1 Appel function shows up in the spectral analysis of the corresponding transition kernel. Independently of any probabilistic consideration a certain multivariable version of the Gelfand-Aomoto hypergeometric function was considered in papers by H. Mizukawa and H. Tanaka. These authors and others such as P. Iliev and P. Tertwilliger treat the twodimensional version of the Hoare-Rahman work from a Lie-theoretic point of view. P. Iliev then treats the general n-dimensional case. All of these authors proved several properties of these functions. Here we show that these functions play a crucial role in the spectral analysis of the transition kernel that comes from pushing the work of Hoare-Rahman to the multivariable case. The methods employed here to prove this as well as several properties of these functions are completely different to those used by the authors mentioned above.

show abstract

“…For general values of n, the author shows that such orthogonal polynomials appear as the zonal spherical functions of Gel'fand pairs of complex reflection groups [3]. In general, the author and H. Tanaka give the orthogonality relation of φ A (x; m)s by using the character algebras [4]. R.C.…”

Section: Introductionmentioning

confidence: 99%