Big q Legendre polynomials - Alchetron, the free social encyclopedia

In mathematics, the big q-Legendre polynomials are an orthogonal family of polynomials defined in terms of Heine's basic hypergeometric series as

P n ( x ; c ; q ) = 3 ϕ 2 ( q − n , q n + 1 , x ; q , c q ; q , q ) .

They obey the orthogonality relation

∫ c q q P m ( x ; c ; q ) P n ( x ; c ; q ) d x = q ( 1 − c ) 1 − q 1 − q 2 n + 1 ( c − 1 q ; q ) n ( c q ; q ) n ( − c q 2 ) n q ( n 2 ) δ m n

and have the limiting behavior

lim q → 1 P n ( x ; 0 ; q ) = P n ( 2 x − 1 )

where P n is the n th Legendre polynomial.

References

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