read `EKHAD.txt`:

#beta=b



GS1:=(a,b,c)->int(2*GAMMA(2*b)/GAMMA(b)^2*(c-(a+1)^2)*(c-a*(a+t))^b*(c-(a+1)*(a+t))^(b-1)/(c-(a+t)^2)^(2*b)*t^b*(1-t)^(b-1),t=0..1):

GS2:=(a,b,c)->int(GAMMA(2*b)/GAMMA(b)^2*(c-(a+1)^2)*(c-a*(a+t))^(b-1)*(c-(a+1)*(a+t))^(b-1)/(c-(a+t)^2)^(2*b)*t^(b-1)*(1-t)^(b-1)*(c-(a-t)*(a+t)),t=0..1):


ope1:=AZd(2*GAMMA(2*b)/GAMMA(b)^2*(c-a(a+1)^2)*(c-a*(a+t))^b*(c-(a+1)*(a+t))^(b-1)/(c-(a+t)^2)^(2*b)*t^b*(1-t)^(b-1),t,b,B)[1]:

ope2:=AZd(GAMMA(2*b)/GAMMA(b)^2*(c-(a+1)^2)*(c-a*(a+t))^(b-1)*(c-(a+1)*(a+t))^(b-1)/(c-(a+t)^2)^(2*b)*t^(b-1)*(1-t)^(b-1)*(c-(a-t)*(a+t)),t,b,B)[1]:


print(`Algorithmic Proofs of Two Curious Integral Identities of  George Gasper and Michael Schlosser`):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):

print(`Gasper and Schlosser discovered and proved (and another human proof was given by Mizan Rahman) `):
print(``):
print(`See:Some curious q-series expansions and beta integral evaluations, Ramanujan J. 13 (1-3) (2007), 229-242`):
print(``):
print(` https://www.mat.univie.ac.at/~schlosse/GScbeta.html `):
print(``):
print(`the following two elegant identities`):
print(``):
print(Int(2*GAMMA(2*b)/GAMMA(b)^2*(c-(a+1)^2)*(c-a*(a+t))^b*(c-(a+1)*(a+t))^(b-1)/(c-(a+t)^2)^(2*b)*t^b*(1-t)^(b-1),t=0..1)=1):

print(``):

print(Int(GAMMA(2*b)/GAMMA(b)^2*(c-(a+1)^2)*(c-a*(a+t))^(b-1)*(c-(a+1)*(a+t))^(b-1)/(c-(a+t)^2)^(2*b)*t^(b-1)*(1-t)^(b-1)*(c-(a-t)*(a+t)),t=0..1)=1):

print(``):
print(`Let's call these two integrals G1(b) and G2(b) respectively`):
print(``):
print(`Using the amazing Almkvist-Zeilberger algorithm , implemented in procedure AZd, in Doron Zeilberger's Maple package EKHAD.txt`):
print(``):
print(`available from http://sites.math.rutgers.edu/~zeilberg/tokhniot/EKHAD.txt `):
print(``):
print(`it is proved rigorously that both satisfy the THIRD-ORDER recurrences`):
print(``):
print(add(coeff(ope1,B,i)*G1(b+i),i=0..degree(ope1,B))=0 ):
print(``):
print(``):
print(add(coeff(ope2,B,i)*G2(b+i),i=0..degree(ope2,B))=0 ):
print(``):
print(`The proof certificates are omitted here, but can be gotten by the reader by  typing`):
print(``):
print(`AZd(2*GAMMA(2*b)/GAMMA(b)^2*(c-a(a+1)^2)*(c-a*(a+t))^b*(c-(a+1)*(a+t))^(b-1)/(c-(a+t)^2)^(2*b)*t^b*(1-t)^(b-1),t,b,B);`):
print(``):
print(`AZd(GAMMA(2*b)/GAMMA(b)^2*(c-(a+1)^2)*(c-a*(a+t))^(b-1)*(c-(a+1)*(a+t))^(b-1)/(c-(a+t)^2)^(2*b)*t^(b-1)*(1-t)^(b-1)*(c-(a-t)*(a+t)),t,b,B);`):
print(``):
print(`in a Maple session where EKHAD.txt has been read to. `):
print(``):
print(`On the other hand, the discrete function that is IDENTICALLY 1 also satisfies these recurrences. Indeed, replacing G1(b) and G2(b) with 1`):
print(``):
print(`and simplifying the algebra, we get that the left sides equal`):
print(``):
print(normal(subs(B=1,ope1))):
print(``):
print(normal(subs(B=1,ope2))):
print(``):
print(`It remains to prove the identities for b=0,b=1,b=2, but these are routine Calculus 1 integrals that are left to the readers.`):
print(``):
print(`This proves the Gasper-Schlosser identities for all NON-NEGATIVE integers, b. By the usual  "analytic continuation" and/or Carlson's theorem,`):
print(``):
print(`this is true for all b where the integrals makes sense. `):
print(``):
print(`----------------------------------`):
print(`This ends this article that took`, time(), `seconds to generate`):

quit: