Jackson's Difference Fan

Author

Eric W. Weisstein
February 17, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Combinatorics/JacksonsDifferenceFan.nb.

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

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

Code

JacksonJ[a[k], {k, h}, n] == (∑_(k=0)^n)SuperscriptBox[(-h),n-k]Binomial[n,k]a[k] /;
n ∈ Integers ∧ n >= 0

InverseJacksonJ[b[k], {k, h}, n] == (∑_(k=0)^n)SuperscriptBox[h,n-k]Binomial[n,k]b[k] /;
n ∈ Integers ∧ n >= 0

BinomialJ[a[k], k, n] == (∑_(k=0)^n)SuperscriptBox[(-1),n-k]Binomial[n,k]a[k] /;
n ∈ Integers ∧ n >= 0

InverseBinomialJ[b[k], k, n] == (∑_(k=0)^n)Binomial[n,k]b[k] /; n ∈ Integers ∧ n >= 0

JacksonJ[InverseJacksonJ[b[k], {k, h}, n], {k, h}, n] == (∑_(k=0)^n)SuperscriptBox[(-h),n-k]Binomial[n,k](∑_(j=0)^k)SuperscriptBox[h,k-j]Binomial[k,j]b[j] == (∑_(j=0)^n)(∑_(k=j)^n)SuperscriptBox[(-h),n-k]Binomial[n,k]SuperscriptBox[h,k-j]Binomial[k,j]b[j] == (∑_(j=0)^n)(((-h))^(-j+n)) b[j] Binomial[n,j] HypergeometricPFQ[{j-n},{},1] == b[n]

Checks

RSolve[(∑_(k=0)^n)(-h)^(n-k)Binomial[n,k]a[k] ==b[n], b, n]

Sum[(-h)^(n-k)*Binomial[n,k]*h^(k-j)*Binomial[k,j]*b[j],{k,j,n}]

FullSimplify[%]

Table[Sum[(-h)^(-j+n)*b[j]*Binomial[n,j]*HypergeometricPFQ[{j-n},{},1],{j,0,n}],{n,0,6}]