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Stieltjes-Wigert Polynomial


Orthogonal polynomials associated with weighting function

w(x)=pi^(-1/2)kexp(-k^2ln^2x)
(1)
=pi^(-1/2)kx^(-k^2lnx)
(2)

for x in (0,infty) and k>0. Defining

 q=exp[-(2k^2)^(-1)],
(3)

then

p_0(x)=q^(1/4)
(4)
p_n(x)=((-1)^nq^(n/2+1/4))/(sqrt((q;q)_n))sum_(nu=0)^(n)[n; nu]q^(nu^2)(-sqrt(q)x)^nu,
(5)

where (q;a)_n is a q-Pochhammer symbol and [n; nu] is a q-binomial coefficient.


See also

q-Binomial Coefficient

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References

Szegö, G. Orthogonal Polynomials, 4th ed. Providence, RI: Amer. Math. Soc., p. 33, 1975.

Referenced on Wolfram|Alpha

Stieltjes-Wigert Polynomial

Cite this as:

Weisstein, Eric W. "Stieltjes-Wigert Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Stieltjes-WigertPolynomial.html

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