Artifact 8b7e1e8f5bab8dba61e1f4f868652f1a5e73395b9a5a377c910f2066a79a4fd2:
- Executable file
r36/XMPL/ZEILBERG.TST
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2011-09-02 18:13:33
on branch master
— Some historical releases purely for archival purposes
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% Tests of the ZEILBERG package. % Authors: Gregor Stoelting, Wolfram Koepf (koepf@zib-berlin.de) load zeilberg; % on time; % 1) Successful summations by the Gosper algorithm % R. W. Gosper, Jr.: % Decision procedure for indefinite hypergeometric summation, % Proc. Nat. Acad. Sci. USA 75 (1978), 40-42. gosper(k,k); gosper(k^2,k); gosper(k^3,k); gosper(k^4,k); gosper(k^5,k); % gosper(k^20,k); gosper((6*k+3)/(4*k^4+8*k^3+8*k^2+4*k+3),k); % gosper(2^k*(k^3-3*k^2-3*k-1)/(k^3*(k+1)^3),k); gosper(x*k,k); gosper(k*x^k,k); gosper(k*factorial(k),k); gosper(1/(k^2-1),k); gosper((1+2*k)/((1+k^2)*(k^2+2*k+2)),k); gosper((k^2+4*k+1)*factorial(k),k); gosper((4*k-3)*factorial(2*k-2)/factorial(k-1),k); gosper(gamma(k+n+2)*n/((k+n+1)*gamma(k+2)*gamma(n+1)),k); gosper((k+n)*factorial(k+n),k); gosper((3*(1+2*k))/((1+k^2)*(2+2*k+k^2)),k); % gosper((-25+15*k+18*k^2-2*k^3-k^4)/ % (-23+479*k+613*k^2+137*k^3+53*k^4+5*k^5+k^6),k); % gosper(3^k*(2*k^4+4*k^3-7*k^2-k-4)/(k*(k+1)*(k^2+1)*((k+1)^2+1)),k); gosper(3^k*(4*k^2-2*k-3)/((2*k+3)*(2*k+1)*(k+1)*k),k); % gosper(2^k*(2*k^3+3*k^2-20*k-15)/ % ((2*k+3)*(2*k+1)*(k+5)*(k+4)*(k+1)*k),k); % gosper(-2^k*((k+11)^2*(k+1)^2-2*(k+10)^2*k^2)/ % ((k+11)^2*(k+10)^2*(k+1)^2*k^2),k); % gosper(-2^k*((k+6)^2*(k+1)^2-2*(k+5)^2*k^2)/ % ((k+6)^2*(k+5)^2*(k+1)^2*k^2),k); % gosper(2^k*(k^4-14*k^2-24*k-9)/(k^2*(k+1)^2*(k+2)^2*(k+3)^2),k); % gosper(((k^2-k-n^2-n-2)*gamma(k+n+2)*gamma(n+1))/ % (2*(-1)^k*2^k*(k+n+1)*gamma(-(k-n-1))*gamma(k+2)),k); % gosper(1/(k+1)*binomial(2*k,k)/(n-k+1)*binomial(2*n-2*k,n-k),k); gosper(3^k*(4*k^2+2*a*k-4*k-2-a)/((2*k+2+a)*(2*k+a)*(k+1)*k),k); gosper(2^k*(k^2-2*k-1)/(k^2*(k+1)^2),k); gosper((3*k^2+3*k+1)/(k^3*(k+1)^3),k); gosper((6*k+3)/(4*k^4+8*k^3+8*k^2+4*k+3),k); gosper(-(k^2+3*k+3)/(k^4+2*k^3-3*k^2-4*k+2),k); gosper(k^2*4^k/((k+1)*(k+2)),k); gosper((2*k-1)^3,k); gosper(3*k^2+3*k+1,k); gosper((k^2+4*k+2)*2^k,k); % gosper(2^k*(k^3-3*k^2-3*k-1)/(k^3*(k+1)^3),k); gosper(k*n^k,k); % gosper(3^k*(2*k^3+k^2+3*k+6)/((k^2+2)*(k^2+2*k+3)),k); % gosper(4*(1-k)*(k^2-2*k-1)/(k^2*(k+1)^2*(k-2)^2*(k-3)^2),k); % gosper(2^k*(k^4-14*k^2-24*k-9)/(k^2*(k+1)^2*(k+2)^2*(k+3)^2),k); gosper((1+k)/(1-k)+2/k,k); gosper(1/(k*(k+1)),k); gosper(1/(k*(k+2)),k); gosper(1/(k*(k+10)),k); % gosper(1/(k*(k+30)),k); gosper(1/(k*(k+1)*(k+2)),k); gosper(1/(k*(k+1)*(k+2)*(k+3)*(k+4)*(k+5)*(k+6)*(k+7)*(k+8)*(k+9)* (k+10)),k); gosper(pochhammer(k-n,n),k); gosper((a+k-1)*pochhammer(a,k),k); gosper((a-k-1)/pochhammer(a-k,k),k); gosper(binomial(k,n),k); gosper(k*binomial(k,n),k); % gosper(k^10*binomial(k,n),k); gosper(1/binomial(k,n),k); gosper(k/binomial(k,n),k); % gosper(k^10/binomial(k,n),k); gosper(binomial(k-n,k),k); gosper((-1)^k*binomial(n,k),k); gosper((-1)^k/binomial(n,k),k); gosper((-1)^(k+1)*(4*k+1)*factorial(2*k)/ (factorial(k)*4^k*(2*k-1)*factorial(k+1)),k); % term:=3^k*(3*k^2+2*a*k-4*k-2-a)/((2*k+2+a)*(2*k+a)*(k+1)*k)$ % term:=sub(k=k+3,term)-term$ % gosper(term,k); % clear(term); % 2) Examples for the Wilf-Zeilberger method: % H. S. Wilf and D. Zeilberger: % Rational functions certify combinatorial identities. % J. Amer. Math. Soc. 3, 1990, 147-158. % Binomial theorem summand:=binomial(n,k)/2^n$ gosper(sub(n=n+1,summand)-summand,k); % Vandermonde summand:=binomial(n,k)^2/binomial(2*n,n)$ gosper(sub(n=n+1,summand)-summand,k); % Gauss % summand:=factorial(n+k)*factorial(b+k)*factorial(c-n-1)*factorial(c-b-1) % /(factorial(n-1)*factorial(c-n-b-1)*factorial(k+1)*factorial(c+k)* % factorial(b-1))$ % gosper(sub(n=n+1,summand)-summand,k); % Kummer % summand:=(-1)^(n+k)*factorial(2*n+c-1)*factorial(n)*factorial(n+c-1)/( % factorial(2*n+c-1-k)*factorial(2*n-k)*factorial(c+k-1)*factorial(k))$ % gosper(sub(n=n+1,summand)-summand,k); % Saalschuetz % summand:=factorial(a+k-1)*factorial(b+k-1)*factorial(n)* % factorial(n+c-a-b-k-1)*factorial(n+c-1)/(factorial(k)*factorial(n-k)* % factorial(k+c-1)*factorial(n+c-a-1)*factorial(n+c-b-1))$ % gosper(sub(n=n+1,summand)-summand,k); % Dixon % summand:=(-1)^k*binomial(n+b,n+k)*binomial(n+c,c+k)*binomial(b+c,b+k)* % factorial(n)*factorial(b)*factorial(c)/factorial(n+b+c)$ % gosper(sub(n=n+1,summand)-summand,k); % 3) Results from Gosper's original work % R. W. Gosper, Jr.: % Decision procedure for indefinite hypergeometric summation, % Proc. Nat. Acad. Sci. USA 75 (1978), 40-42. % ff(k)=product(a+b*j+c*j^2,j,1,k); % gg(k)=product(e+b*j+c*j^2,j,1,k); operator ff,gg; let {ff(~k+~m) => ff(k+m-1)*(c*(k+m)^2+b*(k+m)+a) when (fixp(m) and m>0), ff(~k+~m) => ff(k+m+1)/(c*(k+m+1)^2+b*(k+m+1)+a) when (fixp(m) and m<0)}; let {gg(~k+~m) => gg(k+m-1)*(c*(k+m)^2+b*(k+m)+e) when (fixp(m) and m>0), gg(~k+~m) => gg(k+m+1)/(c*(k+m+1)^2+b*(k+m+1)+e) when (fixp(m) and m<0)}; gosper(ff(k-1)/gg(k),k); % gosper(ff(k-1)/gg(k+1),k); % gosper(ff(k-1)/gg(k+2),k); % ff(k)=product(a+b*j+c*j^2+d*j^3,j,1,k); % gg(k)=product(e+b*j+c*j^2+d*j^3,j,1,k); let { ff(~k+~m) => ff(k+m-1)*(d*(k+m)^3+c*(k+m)^2+b*(k+m)+a) when (fixp(m) and m>0), ff(~k+~m) => ff(k+m+1)/(d*(k+m+1)^3+c*(k+m+1)^2+b*(k+m+1)+a) when (fixp(m) and m<0)}; let { gg(~k+~m) => gg(k+m-1)*(d*(k+m)^3+c*(k+m)^2+b*(k+m)+e) when (fixp(m) and m>0), gg(~k+~m) => gg(k+m+1)/(d*(k+m+1)^3+c*(k+m+1)^2+b*(k+m+1)+e) when (fixp(m) and m<0)}; gosper(ff(k-1)/gg(k),k); gosper(ff(k-1)/gg(k+1),k); % Decision: no closed form solution exists % ff(k)=product(a+b*j+c*j^2+d*j^3+e*j^4,j,1,k); % gg(k)=product(f+b*j+c*j^2+d*j^3+e*j^4,j,1,k); let { ff(~k+~m) => ff(k+m-1)*(e*(k+m)^4+d*(k+m)^3+c*(k+m)^2+b*(k+m)+a) when (fixp(m) and m>0), ff(~k+~m) => ff(k+m+1)/(e*(k+m+1)^4+d*(k+m+1)^3+c*(k+m+1)^2+b*(k+m+1)+a) when (fixp(m) and m<0)}; let { gg(~k+~m) => gg(k+m-1)*(e*(k+m)^4+d*(k+m)^3+c*(k+m)^2+b*(k+m)+f) when (fixp(m) and m>0), gg(~k+~m) => gg(k+m+1)/(e*(k+m+1)^4+d*(k+m+1)^3+c*(k+m+1)^2+b*(k+m+1)+f) when (fixp(m) and m<0)}; gosper(ff(k-1)/gg(k),k); % ff=product(j^3+b*j^2+c*j+(2*c-4*b+8),j,1,k); % gg=product(j^3+b*j^2+c*j,j,1,k) let { ff(~k+~m) => ff(k+m-1)*((k+m)^3+c*(k+m)^2+b*(k+m)+(2*c-4*b+8)) when (fixp(m) and m>0), ff(~k+~m) => ff(k+m+1)/((k+m+1)^2+c*(k+m+1)^2+b*(k+m+1)+(2*c-4*b+8)) when (fixp(m) and m<0)}; let { gg(~k+~m) => gg(k+m-1)*((k+m)^3+c*(k+m)^2+b*(k+m)+1) when (fixp(m) and m>0), gg(~k+~m) => gg(k+m+1)/((k+m+1)^2+c*(k+m+1)^2+b*(k+m+1)+1) when (fixp(m) and m<0)}; gosper(ff(k-1)/gg(k),k); clear(ff,gg); % 4) Examples for which gosper decides that no hypergeometric term % antidifference exists gosper(factorial(k),k); gosper(factorial(2*k)/(factorial(k)*factorial(k+1)),k); % gosper(1/(factorial(k)*(k^4+k^2+1)),k); gosper(binomial(A,k),k); gosper(1/k,k); gosper((1+k)/(1-k),k); % gosper(3^k*(3*k^2+2*a*k-4*k-2-a)/((2*k+2+a)*(2*k+a)*(k+1)*k),k); gosper(factorial(k+n)*factorial(n)/ ((-1)^k*factorial(n-k)*factorial(k)*2^k),k); gosper(1/(k*(k+1/2)),k); gosper(pochhammer(a,k),k); gosper(binomial(n,k),k); % 5) Finding recurrence equations for definite sums % D. Zeilberger, % A fast algorithm for proving terminating hypergeometric identities, % Discrete Math. 80 (1990), 207-211. sumrecursion(binomial(n,k),k,n); sumrecursion(k*binomial(n,k),k,n); % sumrecursion( % (-1)^k*binomial(2*n,k)*binomial(2*k,k)*binomial(4*n-2*k,2*n-k),k,n); sumrecursion(binomial(n,k)^2,k,n); sumrecursion(binomial(n,k)^2/binomial(2*n,n),k,n); % sumrecursion((-1)^k*binomial(n,k)^2,k,n); % Gauss sumrecursion( factorial(n+k)*factorial(b+k)*factorial(c-n-1)*factorial(c-b-1),k,n); sumrecursion( pochhammer(a,k)*pochhammer(b,k)/(factorial(k)*pochhammer(c,k)),k,a); % Kummer sumrecursion((-1)^(n+k)*factorial(2*n+c-1)*factorial(n)*factorial(n+c-1) /(factorial(2*n+c-1-k)*factorial(2*n-k)*factorial(c+k-1)* factorial(k)),k,n); sumrecursion((-1)^k/( factorial(2*n+c-1-k)*factorial(2*n-k)*factorial(c+k-1)* factorial(k)),k,n); % Saalschuetz % sumrecursion(factorial(a+k-1)*factorial(b+k-1)*factorial(n)* % factorial(n+c-a-b-k-1)*factorial(n+c-1)/ % (factorial(k)*factorial(n-k)*factorial(k+c-1)* % factorial(n+c-a-1)*factorial(n+c-b-1)),k,n); sumrecursion(factorial(a+k-1)*factorial(b+k-1)*factorial(n+c-a-b-k-1)/( factorial(k)*factorial(n-k)*factorial(k+c-1)),k,n); % Dixon % sumrecursion((-1)^k*binomial(n+b,n+k)*binomial(n+c,c+k)* % binomial(b+c,b+k)* % factorial(n)*factorial(b)*factorial(c)/factorial(n+b+c),k,n); sumrecursion((-1)^k*binomial(n+b,n+k)*binomial(n+c,c+k)* binomial(b+c,b+k),k,n); sumrecursion((-1)^(k-n)*binomial(2*n,k)^3,k,n); sumrecursion( (-1)^(k-n)*binomial(2*n,k)^3/(binomial(3*n,n)*binomial(2*n,n)),k,n); % Clausen % summand:=factorial(a+k-1)*factorial(b+k-1)/ % (factorial(k)*factorial(-1/2+a+b+k))*factorial(a+n-k-1)* % factorial(b+n-k-1)/(factorial(n-k)*factorial(-1/2+a+b+n-k))$ % sumrecursion(summand,k,n); % Dougall % summand:= % pochhammer(d,k)*pochhammer(1+d/2,k)*pochhammer(d+b-a,k)* % pochhammer(d+c-a,k)* % pochhammer(1+a-b-c,k)*pochhammer(n+a,k)*pochhammer(-n,k)/ % (factorial(k)* % pochhammer(d/2,k)*pochhammer(1+a-b,k)*pochhammer(1+a-c,k)* % pochhammer(b+c+d-a,k)*pochhammer(1+d-a-n,k)*pochhammer(1+d+n,k))$ % sumrecursion(summand,k,n); % Apery sumrecursion(binomial(n,k)^2*binomial(n+k,k)^2,k,n); % sumrecursion(4*(-1)^k*binomial(m-1,k)*binomial(2*m-1,2*k)/ % binomial(4*m-1,4*k)*(4*m^2+16*k^2-16*k*m+16*k-6*m+3)/ % ((4*m-4*k-3)*(4*m-4*k-1)),k,m); sumrecursion((-1)^k*binomial(n,k)*binomial(k,n),k,n); sumrecursion((-1)^k*binomial(n,k)*binomial(2*k,n),k,n); sumrecursion((-1)^k*binomial(n,k)*binomial(k,j)^2,k,n); sumrecursion(binomial(n,k)*binomial(a,k),k,n); sumrecursion((3*k-2*n)*binomial(n,k)^2*binomial(2*k,k),k,n); sumrecursion(binomial(n-k,k),k,n); sumrecursion(binomial(n,k)*binomial(n+k,k),k,n); % sumrecursion(binomial(n+k,m+2*k)*binomial(2*k,k)*(-1)^k/(k+1),k,n); sumrecursion((-1)^k*binomial(n-k,k)*binomial(n-2*k,m-k),k,n); % sumrecursion((-1)^k*binomial(n-k,k)*binomial(n-2*k,m-k),k,m); sumrecursion(binomial(n+k,2*k)*2^(n-k),k,n); sumrecursion(binomial(n,k)*binomial(2*k,k)*(-1/4)^k,k,n); % sumrecursion(binomial(n,i)*binomial(2*n,n-i),i,n); sumrecursion((-1)^k*binomial(n,k)*binomial(2*k,k)*4^(n-k),k,n); sumrecursion((-1)^k*binomial(n,k)/binomial(k+a,k),k,n); % sumrecursion((-1)^k*binomial(n,k)/binomial(k+a,k),k,a); sumrecursion((-1)^(n-k)*binomial(2*n,k)^2,k,n); sumrecursion(factorial(a+k)*factorial(b+k)*factorial(n+c-a-b-k-1)/( factorial(k+1)*factorial(n-k)*factorial(k+c)),k,a); % sumrecursion(factorial(a+k)*factorial(b+k)*factorial(n+c-a-b-k-1)/( % factorial(k+1)*factorial(n-k)*factorial(k+c)),k,b); % sumrecursion(factorial(a+k)*factorial(b+k)*factorial(n+c-a-b-k-1)/( % factorial(k+1)*factorial(n-k)*factorial(k+c)),k,c); sumrecursion(binomial(2*n+1,2*p+2*k+1)*binomial(p+k,k),k,n); % sumrecursion(binomial(2*n+1,2*p+2*k+1)*binomial(p+k,k),k,p); sumrecursion(binomial(r,m)*binomial(s,t-m),m,r); % sumrecursion(binomial(r,m)*binomial(s,t-m),m,s); % sumrecursion(binomial(r,m)*binomial(s,t-m),m,t); sumrecursion(binomial(2*n+1,2*k)*binomial(m+k,2*n),k,n); % sumrecursion(binomial(2*n+1,2*k)*binomial(m+k,2*n),k,m); sumrecursion(binomial(n,k)*binomial(k,j)*x^j,k,n); % sumrecursion(binomial(n,k)*binomial(k,j)*x^j,k,j); % sumrecursion(binomial(n,k)*binomial(k,j)*x^k,k,n); sumrecursion(x*binomial(n+k,2*k)*((x^2-1)/4)^(n-k),k,n); sumrecursion(binomial(n+k-1,2*k-1)*(x-1)^(2*k)*x^(n-k)/k,k,n); sumrecursion( 1/(k+1)*binomial(2*k,k)/(n-k+1)*binomial(2*n-2*k,n-k),k,n); sumrecursion(binomial(m,r)*binomial(n-r,n-r-q)*(t-1)^r,r,m); % sumrecursion(binomial(m,r)*binomial(n-r,n-r-q)*(t-1)^r,r,n); % sumrecursion(binomial(m,r)*binomial(n-r,n-r-q)*(t-1)^r,r,q); % sumrecursion(binomial(m,r)*binomial(n-r,n-r-q)*(t-1)^r,r,r); sumrecursion(pochhammer(-n/2,k)*pochhammer(-n/2+1/2,k)/ (factorial(k)*pochhammer(b+1/2,k)),k,n); % Watson % sumrecursion(pochhammer(a,k)*pochhammer(b,k)*pochhammer(c,k)/( % factorial(k)*pochhammer(1/2*(a+b+1),k)*pochhammer(2*c,k)),k,c); % sumrecursion(pochhammer(-m,j)*pochhammer(m+2*k+2,j)*pochhammer(k+1/2,j)/ % (factorial(j)*pochhammer(k+3/2,j)*pochhammer(2*k+1,j)),j,k); sumrecursion((-1)^k*binomial(n,k)^3,k,n); % sumrecursion(pochhammer(-n,k)*pochhammer(n+2*a,k)*pochhammer(a,k)/( % factorial(k)*pochhammer(2*a/2,k)*pochhammer((2*a+1)/2,k))*(2/4)^k,k,n); % sumrecursion(pochhammer(-n,k)*pochhammer(n+4*a,k)*pochhammer(a,k)/( % factorial(k)*pochhammer(4*a/2,k)*pochhammer((4*a+1)/2,k))*(4/4)^k,k,n); % sumrecursion(binomial(n+k+1,n-k)*pochhammer(-n+k,j)*pochhammer(k+1/2,j)* % pochhammer(n+k+2,j)/(factorial(j)*pochhammer(k+3/2,j)* % pochhammer(2*k+1,j)),j,n); % sumrecursion(pochhammer(-m,j)*pochhammer(m+2*k+2,j)* % pochhammer(k+1/2,j)/( % factorial(j)*pochhammer(k+3/2,j)*pochhammer(2*k+1,j)),j,m); % sumrecursion(binomial(n+k+1,n-k)*pochhammer(-n+k,j)* % pochhammer(k+1/2,j)* % pochhammer(n+k+2,j)/(factorial(j)*pochhammer(k+3/2,j)* % pochhammer(2*k+1,j)), % j,k); % sumrecursion(pochhammer(a+b+c-n,j+l)*pochhammer(a+b-n/2,j+l)/ % (factorial(j)*factorial(l)*pochhammer(a-n/2+1,j)* % pochhammer(b-n/2+1,l)),j,a); % sumrecursion(pochhammer(a+b+c-n,j+l)*pochhammer(a+b-n/2,j+l)/ % (factorial(j)*factorial(l)*pochhammer(a-n/2+1,j)* % pochhammer(b-n/2+1,l)),j,b); sumrecursion(pochhammer(a+b+c-n,j+l)*pochhammer(a+b-n/2,j+l)/ (factorial(j)*factorial(l)*pochhammer(a-n/2+1,j)* pochhammer(b-n/2+1,l)),j,c); % sumrecursion( % (-1)^(a+b+c)*gamma(a+b+c-d/2)*gamma(d/2-c)*gamma(a+c-d/2)* % gamma(b+c-d/2)/ % (gamma(a)*gamma(b)*gamma(d/2)*gamma(a+b+2*c-d)*(m^2)^(a+b+c-d))* % pochhammer(a+b+c-d,k)*pochhammer(a+c-d/2,k)/ % (pochhammer(a+b+2*c-d,k)*factorial(k)),k,a); % sumrecursion( % (-1)^(a+b+c)*gamma(a+b+c-d/2)*gamma(d/2-c)*gamma(a+c-d/2)* % gamma(b+c-d/2)/ % (gamma(a)*gamma(b)*gamma(d/2)*gamma(a+b+2*c-d)*(m^2)^(a+b+c-d))* % pochhammer(a+b+c-d,k)*pochhammer(a+c-d/2,k)/ % (pochhammer(a+b+2*c-d,k)*factorial(k)),k,b); % sumrecursion( % (-1)^(a+b+c)*gamma(a+b+c-d/2)*gamma(d/2-c)*gamma(a+c-d/2)* % gamma(b+c-d/2)/ % (gamma(a)*gamma(b)*gamma(d/2)*gamma(a+b+2*c-d)*(m^2)^(a+b+c-d))* % pochhammer(a+b+c-d,k)*pochhammer(a+c-d/2,k)/ % (pochhammer(a+b+2*c-d,k)*factorial(k)),k,c); % sumrecursion(pochhammer(-n,k)*pochhammer(n+3*a,k)*pochhammer(a,k)/( % factorial(k)*pochhammer(3*a/2,k)* % pochhammer((3*a+1)/2,k))*(3/4)^k,k,n); % summand:=k*(-1)^j*pochhammer(2*k+j+1,j)*pochhammer(2*k+2*j+2,n-k-j)/( % factorial(k+j)*factorial(j)*factorial(n-k-j))*exp(-(j+k)*t)$ % summand:=k*(-1)^j*pochhammer(2*k+j+1,j)*pochhammer(2*k+2*j+2,n-k-j)/( % (k+j)*factorial(j)*factorial(n-k-j))*exp(-(j+k)*t)$ % sumrecursion(summand,j,n); clear(summand); % 6) Finding recurrence equations for hypergeometric functions % Koornwinder, T. H.: % On Zeilberger's algorithm and % its q-analogue: a rigorous description. % J. of Comput. and Appl. Math. 48, 1993, 91-111. % Gauss hyperrecursion({a,b},{c},1,a); % Dougall % hyperrecursion({d,1+d/2,d+b-a,d+c-a,1+a-b-c,n+a,-n}, % {d/2,1+a-b,1+a-c,b+c+d-a,1+d-a-n,1+d+n},1,n); % Baxter % hyperrecursion({-n,-n-1,-n-2},{2,3},-1,n); % Krawtchouk polynomials % krawtchoukterm := % (-1)^n*p^n*binomial(NN,n)*hyperterm({-n,-x},{-NN},1/p,k)$ % sumrecursion(krawtchoukterm,k,n); % sumrecursion(krawtchoukterm,k,x); % sumrecursion(krawtchoukterm,k,NN); % clear(krawtchoukterm); % hyperrecursion({-n,b,c+4},{b+1,c},1,n); hyperrecursion({-n,b,c+1,d+1},{b+1,c,d},1,n); % 7) Extended versions of Gosper's and Zeilberger's algorithms % Koepf, W.: % Algorithms for the indefinite and definite summation. % Konrad-Zuse-Zentrum Berlin (ZIB), Preprint SC 94-33, 1994. % extended Gosper algorithm extended_gosper(k*factorial(k/7),k,7); extended_gosper(k*factorial(k/2),k,2); extended_gosper(k*factorial(k/2),k); extended_gosper(binomial(k/2,n),k); extended_gosper(binomial(n,k/2)-binomial(n,k/2-1),k); % extended Zeilberger algorithm % extended_sumrecursion(binomial(n,k)*binomial(k/2,n),k,n,1,2); sumrecursion(binomial(n,k)*binomial(k/2,n),k,n); extended_sumrecursion(binomial(n/2,k),k,n,2,1); sumrecursion(binomial(n/2,k),k,n); % sumrecursion(hyperterm({a,b,a+1/2-b,1+2*a/3,-n}, % {2*a+1-2*b,2*b,2/3*a,1+a+n/2},4,k)/ % (factorial(n)*2^(-n)/factorial(n/2))/ % hyperterm({a+1,1},{a-b+1,b+1/2},1,n/2),k,n); % Watson % sumrecursion(pochhammer(a,k)*pochhammer(b,k)*pochhammer(c,k)/( % factorial(k)*pochhammer(1/2*(a+b+1),k)*pochhammer(2*c,k))/ % (GAMMA(1/2)*GAMMA(1/2+c)*GAMMA(1/2+a/2+b/2)*GAMMA(1/2-a/2-b/2+c))* % GAMMA(1/2+a/2)*GAMMA(1/2+b/2)*GAMMA(1/2-a/2+c)*GAMMA(1/2-b/2+c),k,a); % hyperrecursion({a,b,c},{1/2*(a+b+1),2*c},1,a); % hyperrecursion({a,b,c},{1/2*(a+b+1),2*c},1,b); % 8) Closed form representations of hypergeometric sums % Vandermonde hypersum({-n,b},{c},1,n); % Saalschuetz hypersum({a,b,-n},{c,1+a+b-c-n},1,n); % Kummer hypersum({a,-n},{1+a+n},-1,n); % Dixon hypersum({a,b,-n},{1+a-b,1+a+n},1,n); % Dougall % hypersum({a,1+a/2,b,c,d,1+2*a-b-c-d+n,-n}, % {a/2,1+a-b,1+a-c,1+a-d,1+a-(1+2*a-b-c-d+n),1+a+n},1,n); % Clausen % hypersum({a,b,1/2-a-b-n,-n},{1/2+a+b,1-a-n,1-b-n},1,n); hypersum({a,1+a/2,c,d,-n},{a/2,1+a-c,1+a-d,1+a+n},1,n); hypersum({a,1+a/2,d,-n},{a/2,1+a-d,1+a+n},-1,n); % m-fold case: hypersum({-n,-n,-n},{1,1},1,n); % hypersum({-n,n+3*a,a},{3*a/2,(3*a+1)/2},3/4,n); % 9) Hypergeometric representations sumtohyper(binomial(n,k)^3,k); sumtohyper(binomial(n,k)/2^n-sub(n=n-1,binomial(n,k)/2^n),k); sumtohyper(binomial(k+j-1,k-j)*2*(-1)^(j+1)*factorial(2*j-1)/ factorial(j-1)/factorial(j+1)*x^j,j); % term:=1/(n-1+k)*(1/2-1/2*x)^k/n*binomial(k-n-1,-n-1)*k* % binomial(n-1+k,n-1); % sumtohyper(sub(n=n+1,term)-term,k); end;