In mathematics, the continuous ''q''-Jacobi polynomials ''P''(''x'', ''q''), introduced by , are a family of basic hypergeometric
orthogonal polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonality, orthogonal to each other under some inner product.
The most widely used orthogonal polynomial ...
in the basic
Askey scheme
In mathematics, the Askey scheme is a way of organizing orthogonal polynomials of hypergeometric or basic hypergeometric type into a hierarchy. For the classical orthogonal polynomials discussed in , the Askey scheme was first drawn by and by , ...
. give a detailed list of their properties.
Definition
The polynomials are given in terms of
basic hypergeometric function
In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series.
A series ''x'n'' is called h ...
s and the
q-Pochhammer symbol
In mathematical area of combinatorics, the ''q''-Pochhammer symbol, also called the ''q''-shifted factorial, is the product
(a;q)_n = \prod_^ (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^),
with (a;q)_0 = 1.
It is a ''q''-analog of the Pochhammer symb ...
by
:
References
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*{{cite thesis , last=Sadjang , first=Patrick Njionou , title=Moments of Classical Orthogonal Polynomials , type=Ph.D. , publisher=Universität Kassel , citeseerx=10.1.1.643.3896
Orthogonal polynomials
Q-analogs
Special hypergeometric functions