Saalschütz's Theorem

Author

Eric W. Weisstein
July 9, 2020

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

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

©2020 Wolfram Research, Inc. except for portions noted otherwise

Identities

(Subscript[, 3])(Subscript[F, 2])(a,b,-n;c,a+b-c-n+1;1)==(((Pochhammer[c-a, n]) (Pochhammer[c-b, n]))/((Pochhammer[c, n]) (Pochhammer[-a-b+c, n])))/. n∈(PositiveIntegers) [Bailey]

FullSimplify[Subtract@@{HypergeometricPFQ[{a,b,-n},{c,1+a+b-c-n},1],((Pochhammer[c-a,n]Pochhammer[c-b,n])/(Pochhammer[c,n]Pochhammer[c-a-b,n]))},n∈PositiveIntegers]

FunctionExpand[%,Assumptions->n∈PositiveIntegers]

Table[Subtract@@{HypergeometricPFQ[{a,b,-n},{c,1+a+b-c-n},1],((Pochhammer[-a+c,n] Pochhammer[-b+c,n])/(Pochhammer[c,n] Pochhammer[-a-b+c,n]))}/.{a->RandomComplex[],b->RandomComplex[],c->RandomComplex[],n->RandomInteger[{0,20}]},{1000}]//Chop//Tally

(Subscript[, 3])(Subscript[F, 2])(a,b,c;d,e;1)==(((π^2) (Gamma[d]) (Gamma[e]))/((Gamma[d-a]) (Gamma[e-a]) (Gamma[d-b]) (Gamma[e-b]) (Gamma[d-c]) (Gamma[e-c]) (cos(π a) cos(π b) cos(π c)+cos(π d) cos(π e)))) /; d+e==a+b+c+1 [Gosper]

One of a, b, and c is a nonpositive integer but it is not known which one

d+e==a+b+c+1

id=With[{e=a+b+c+1-d},Subtract@@{
HypergeometricPFQ[{a,b,c},{d,e},1],
((Gamma[d])/(Gamma[d-a]Gamma[d-b]Gamma[d-c]))((Gamma[e])/(Gamma[e-a]Gamma[e-b]Gamma[e-c]))((π^2)/(Cos[π d]Cos[π e]+Cos[π a]Cos[π b]Cos[π c]))}];

a is a nonpositive integer:

Table[id/.{a->-RandomInteger[{0,20}],b->RandomComplex[],c->RandomComplex[],d->RandomComplex[]},{1000}]//Chop//Union

b is a nonpositive integer:

Table[id/.{a->RandomComplex[],b->-RandomInteger[{0,20}],c->RandomComplex[],d->RandomComplex[]},{1000}]//Chop//Union

c is a nonpositive integer:

Table[id/.{a->RandomComplex[],b->RandomComplex[],c->-RandomInteger[{0,20}],d->RandomComplex[]},{1000}]//Chop//Union

a and b are nonpositive integers:

Table[id/.{a->-RandomInteger[{0,20}],b->-RandomInteger[{0,20}],c->RandomComplex[],d->RandomComplex[]},{1000}]//Chop//Union

a and c nonpositive integers:

Table[id/.{a->-RandomInteger[{0,20}],b->RandomComplex[],c->-RandomInteger[{0,20}],d->RandomComplex[]},{1000}]//Chop//Union

b and c nonpositive integers:

Table[id/.{a->RandomComplex[],b->-RandomInteger[{0,20}],c->-RandomInteger[{0,20}],d->RandomComplex[]},{1000}]//Chop//Union

a, b, c nonpositive integers:

Table[id/.{a->-RandomInteger[{0,20}],b->-RandomInteger[{0,20}],c->-RandomInteger[{0,20}],d->RandomComplex[]},{10}]//Chop//Union

(Subscript[, 3])(Subscript[F, 2])(a,b,c;d,e;1)==(((π^2) (Gamma[d]) (Gamma[e]) (-(a+1) (b+1) (c+1)+a b c+d e))/(((Gamma[d-a]) (Gamma[d-b]) (Gamma[d-c])) ((Gamma[e-a]) (Gamma[e-b]) (Gamma[e-c])) (cos(π d) cos(π e)-cos(π a) cos(π b) cos(π c)))) /; d+e==a+b+c+2 [Gosper]

One of a, b, and c is a nonpositive integer but it is not known which one

Equal@@{HypergeometricPFQ[{a,b,c},{d,e},1],((Gamma[d])/(Gamma[d-a]Gamma[d-b]Gamma[d-c]))((Gamma[e])/(Gamma[e-a]Gamma[e-b]Gamma[e-c]))(π^2)((d e-(a+1)(b+1)(c+1)+a b c)/(Cos[π d]Cos[π e]-Cos[π a]Cos[π b]Cos[π c]))}

id=With[{e=a+b+c+2-d},Subtract@@{HypergeometricPFQ[{a,b,c},{d,e},1],((Gamma[d])/(Gamma[d-a]Gamma[d-b]Gamma[d-c]))((Gamma[e])/(Gamma[e-a]Gamma[e-b]Gamma[e-c]))(π^2)((d e-(a+1)(b+1)(c+1)+a b c)/(Cos[π d]Cos[π e]-Cos[π a]Cos[π b]Cos[π c]))}]

Table[id/.{a->-RandomInteger[{0,20}],b->RandomComplex[],c->RandomComplex[],d->RandomComplex[]},{1000}]//Chop//Union

b is a nonpositive integer:

Table[id/.{a->RandomComplex[],b->-RandomInteger[{0,20}],c->RandomComplex[],d->RandomComplex[]},{1000}]//Chop//Tally

c is a nonpositive integer:

Table[id/.{a->RandomComplex[],b->RandomComplex[],c->-RandomInteger[{0,20}],d->RandomComplex[]},{1000}]//Chop//Tally

a and b are nonpositive integers:

Table[id/.{a->-RandomInteger[{0,20}],b->-RandomInteger[{0,20}],c->RandomComplex[],d->RandomComplex[]},{1000}]//Chop//Union

a and c nonpositive integers:

Table[id/.{a->-RandomInteger[{0,20}],b->RandomComplex[],c->-RandomInteger[{0,20}],d->RandomComplex[]},{1000}]//Chop//Union

b and c nonpositive integers:

Table[id/.{a->RandomComplex[],b->-RandomInteger[{0,20}],c->-RandomInteger[{0,20}],d->RandomComplex[]},{1000}]//Chop//Union

a, b, c nonpositive integers:

Table[id/.{a->-RandomInteger[{0,20}],b->-RandomInteger[{0,20}],c->-RandomInteger[{0,20}],d->RandomComplex[]},{10}]//Chop//Union