Gould Polynomial

Author

Eric W. Weisstein
September 7, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Enumeration

FallingFactorial[x_, n_] := (-1)^n Pochhammer[-x, n]

GouldG[x_,n_,{a_,b_}]:=(x/(x-a n))FactorialPower[(x-a n)/b,n]

General

Solve[Exp[a t](Exp[b t]-1)==y,t]

v4.2

Sum[Binomial[(-b+a k)/b,k-1]t^k/k,{k,∞}]/b

v5.0

Sum[Binomial[(-b+a k)/b,k-1]t^k/k,{k,∞}]/b

V7

Sum[Binomial[(-b+a k)/b,k-1]t^k/k,{k,∞}]/b

First Few

FullSimplify[Table[GouldG[x,n,{a,b}],{n,0,4}]//FunctionExpand]//ColumnForm//TraditionalForm

a=-b/2

Factor//@Simplify[Series[Exp[2x ArcSinh[t/2]/b],{t,0,4}]]

Together[Table[GouldG[x,n,{-b/2,b}]//FunctionExpand,{n,0,4}]]//ColumnForm//TraditionalForm

Generating Function

(∑_(k=0)^∞)(((t^k) GouldG(x,k,{a,b}))/(k!))

Sum[((GouldG[x,k,{a,b}])/(k!))(t^k),{k,0,∞}]

(∑_(k=0)^∞)(((t^k) GouldG(x,k,{-(b/2),b}))/(k!))==(∑_(k=0)^∞)((x (t^k) (FactorialPower[(((((b k)/2)+x)/b)), k]))/(k! (((b k)/2)+x)))==exp(((2 x (sinh^(-1))((t/2)))/b))

Sum[((GouldG[x,k,{-b/2,b}])/(k!))(t^k),{k,0,∞}]

Head[Sum[(t^k*x*FactorialPower[((b*k)/2+x)/b,k])/(((b*k)/2+x)*k!),{k,0,Infinity}]]

Subtract@@@Table[{Sum[(t^k*x*FactorialPower[((b*k)/2 + x)/b, k])/(((b*k)/2 + x)*k!), {k, 0, 200}],Exp[2x ArcSinh[t/2]/b]}/.{x->RandomReal[],b->RandomReal[],t->RandomReal[]},{10}]//Chop//Union

Chu-Vandermonde

(∑_(k=0)^n)((x y)/((x-a k) (y-a (n-k)))) (Binomial[((x-a k)/b), k]) (Binomial[((y-a (n-k))/b), n-k])==(((x+y))/(-a n+x+y)) (Binomial[((-a n+x+y)/b), n])

Sum[(x/(x-a k))(y/(y-a(n-k)))Binomial[((x-a k)/b),k]Binomial[((y-a(n-k))/b),n-k],{k,0,n}]

Sum[(x/(x-a k))(y/(y-a(n-k)))Binomial[((x-a k)/b),k]Binomial[((y-a(n-k))/b),n-k],{k,0,n}]//InputForm

Subtract@@@Table[{Sum[(x/(x-a k))(y/(y-a(n-k)))Binomial[((x-a k)/b),k]Binomial[((y-a(n-k))/b),n-k],{k,0,n}],((x+y)/(x+y-a n))Binomial[((x+y-a n)/b),n]},{n,0,10}]//FunctionExpand//Together//Union