Brafman polynomials

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.

References

[a1]	F. Brafman, "Some generating functions for Laguerre and Hermite polynomials" Canadian J. Math. , 9 (1957) pp. 180–187
[a2]	H.W. Gould, A.T. Hopper, "Operational formulas connected with two generalizations of Hermite polynomials" Duke Math. J. , 29 (1962) pp. 51–63
[a3]	D. Mangeron, A.M. Krall, D.L. Fernandez, "Weight functions for some new classes of orthogonal polynomials" R. Acad. Cien. (Madrid) , 77 (1983) pp. 597–607
[a4]	R.M. Shreshtha, "On generalized Brafman polynomials" Comp. R. Acad. Bulgar. Sci. , 32 (1979) pp. 1183–1185