# Brafman polynomials

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Polynomials given by

$$B _ {n} ^ {p} ( a _ {1} \dots a _ {r} ;b _ {1} \dots b _ {s} ;x ) =$$

$$= { {} _ {p + r } F _ {s} } [ \Delta ( p; - n ) , a _ {1} \dots a _ {r} , b _ {1} \dots b _ {s} ;x ] ,$$

where $p$ is a positive integer, $\Delta ( p; - n )$ abbreviates the set of $p$ parameters

$${ \frac{- n }{p} } , - { \frac{( n - 1 ) }{p} } \dots - { \frac{( n - p + 1 ) }{p} } ,$$

and for non-negative integers $r$ and $s$, ${ {} _ {r} F _ {s} }$ denotes the generalized hypergeometric function (cf. also Hypergeometric function), defined by

$${ {} _ {r} F _ {s} } ( a _ {1} \dots a _ {r} ;b _ {1} \dots b _ {s} ;x ) = \sum _ {k = 0 } ^ \infty { \frac{( a _ {1} ) _ {k} \dots ( a _ {r} ) _ {k} x ^ {k} }{( b _ {1} ) _ {k} \dots ( b _ {s} ) _ {k} k! } } .$$

The Brafman polynomials arise in the study of generating functions of orthogonal polynomials, [a1].

There are extensions. H.W. Gould and A.T. Hopper [a2] have considered special cases which sometimes reduce to the Hermite polynomials; see [a4] for a generalization. It is known [a3] that, in general, the Brafman polynomials cannot form an orthogonal set with respect to any weight function.

How to Cite This Entry:
Brafman polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brafman_polynomials&oldid=46144
This article was adapted from an original article by A.M. Krall (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article