zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials. (English) Zbl 0572.33012

This memoir is a detailed exposition of a new family of orthogonal polynomials which has five free parameters and a continuous weight distribution. Indeed, there are cases where this distribution has a finite discrete part in addition. The orthogonality relation is based on a new contour integral. To state the most important results some notation is needed: (throughout q and |q|<1)

(a;q) n := j=0 n-1 (1-aq j ),n=0,1,2,···;e q (a):= j=0 (1-aq j ),
p n (x;a,b,c,d|q):=a -n (ab;q) n (ac;q) n (ad;q) n 4ϕ 3 q -n ,q n-1 abcd,ae iθ ,ae -iθ ;ab,ac,adq,q

where x=cosθ, n=0,1,2,··. (also denoted by p n (x)), a terminating basic hypergeometric series.

The underlying integral depends on five parameters q, a j , (1j4) such that |q|<1 and a j a k q for =0,1,2,···,(1j,k4):

(1/2πi) C e q (z 2 )e q (z -2 ) j=1 4 (e q (a j z)e q (a j /z)) -1 (dz/z)
=2e q (abcd) e q (q) 1j<k4 e q (a j a k ) -1 ,

where C is a closed positively oriented contour consisting of the unit circle deformed so as to separate the sequences of poles converging to zero from the sequences of poles converging to infinity.

It is shown that {p n (x)} is a family of polynomials in x, with degree (p n )=n, and satisfying a three-term recurrence. Further, each p n is symmetric in a,b,c,d. The purely continuous weight distribution occurs for -1<q<1, max(|a|,|b|,|c|,|d|)<1, and a,b,c,d all real or appearing in conjugate pairs: then

-1 1 p n (x)p m (x)w(x)(1-x 2 ) -1/2 dx=0fornm,

where

w(x)=e q (e 2iθ )e q (e -2iθ ) j=1 4 (e q (a j e iθ )e q (a j e -iθ )) -1 ,wherex=cosθ,

and a j (1j4) takes the values a,b,c,d. The integral

-1 1 p n (x) 2 w(x)(1-x 2 ) -1/2 dx

is explicitly found. When any of the parameters a,b,c,d exceed 1, a finite discrete part (explicitly known) is added to the weight distribution.

The power of these results stems from the large number of free parameters. Special choices lead to previously studied families. For example, the case c=-a, b=aq 1/2 =-d gives the Rogers continuous q-ultraspherical polynomials (R. Askey and M. E.-H. Ismail, Studies in pure mathematics, Mem. of P. Turán, 55-78 (1983; Zbl 0532.33006).

Another example comes from the choice a=q α/2+1/4 , c=q 1/2 , b=-q β/2+1/4 , d=bq 1/2 ; this is the family of little q-Jacobi polynomials studied by G. Andrews and R. Askey.

Other special cases are also discussed in the paper, including the example q=0, and a q-analogue of Meixner-Pollaczek polynomials. There is a Rodrigues type formula for p n which depends on a divided difference operator. Also, the connection coefficients between two different families {p n (x;a,b,c,d|q)} and {p n (x;a ' ,b ' ,c ' ,d ' |q)} are given as 5 ϕ 4 -series, and some tractable special cases are discussed. This is a paper of fundamental importance in the theory of orthogonal polynomials in one variable of hypergeometric type.

Reviewer: Ch.Dunkl

MSC:
33D45Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
33D05q-gamma functions, q-beta functions and integrals
39A10Additive difference equations