Wilson Polynomial

Author

Eric W. Weisstein
March 12, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/WilsonPolynomial.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/WilsonPolynomial.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

Version 1

Koekoek, R. and Swarttouw

Wilson[n_,{a_,b_,c_,d_},x_]:=Pochhammer[a+b,n]Pochhammer[a+c,n]Pochhammer[a+d,n]HypergeometricPFQ[{-n,a+b+c+d+n-1,a+I Sqrt[x],a-I Sqrt[x]},{a+b,a+c,a+d},1]

Wilson[0,{a,b,c,d},x]

Wilson[1,{a,b,c,d},(x^2)]

Refine[Table[Wilson[n,{a,b,c,d},x]-Wilson[n,{b,a,c,d},x],{n,0,3}],x>0]//FullSimplify

Version 2

Koepf?

Refine[Wilson[n,{a,b,c,d},-(x^2)],x∈Reals]//TraditionalForm

Wilson2[n_,{a_,b_,c_,d_},x_]:=Pochhammer[a+b,n]Pochhammer[a+c,n]Pochhammer[a+d,n]HypergeometricPFQ[{-n,a+b+c+d+n-1,a-x,a+x},{a+b,a+c,a+d},1]

Wilson2[0,{a,b,c,d},x]

Wilson2[1,{a,b,c,d},x]

Factor[%]

Table[Wilson2[n,{a,b,c,d},x]-Wilson2[n,{b,a,c,d},x],{n,0,3}]//FullSimplify