Artifact 5ba749516e5496b2fef0300e4cb3eb4518e3051ab778ce31f8d8cba73cae97fb:
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r37/packages/sum/zeilberg.rlg
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2011-09-02 18:13:33
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Sat Apr 10 01:33:02 MET DST 1999 REDUCE 3.7, 15-Jan-99 ... 1: 1: 2: 2: 2: 2: 2: 2: 2: 2: 2: 3: 3: % Tests of the ZEILBERG package. % Authors: Gregor Stoelting, Wolfram Koepf (koepf@zib-berlin.de) load_package sum; % 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); (k + 1)*k ----------- 2 gosper(k^2,k); (2*k + 1)*(k + 1)*k --------------------- 6 gosper(k^3,k); 2 2 (k + 1) *k ------------- 4 gosper(k^4,k); 2 (3*k + 3*k - 1)*(2*k + 1)*(k + 1)*k -------------------------------------- 30 gosper(k^5,k); 2 2 2 (2*k + 2*k - 1)*(k + 1) *k ------------------------------ 12 % gosper(k^20,k); gosper((6*k+3)/(4*k^4+8*k^3+8*k^2+4*k+3),k); (k + 2)*k ---------------- 2 2*k + 4*k + 3 % gosper(2^k*(k^3-3*k^2-3*k-1)/(k^3*(k+1)^3),k); gosper(x*k,k); (k + 1)*k*x ------------- 2 gosper(k*x^k,k); k x *((x - 1)*k - 1)*x ---------------------- 2 (x - 1) gosper(k*factorial(k),k); (k + 1)*factorial(k) gosper(1/(k^2-1),k); 2 2*k - 1 ------------- 2*(k + 1)*k gosper((1+2*k)/((1+k^2)*(k^2+2*k+2)),k); (k + 2)*k ------------------ 2 2*(k + 2*k + 2) gosper((k^2+4*k+1)*factorial(k),k); (k + 4)*(k + 1)*factorial(k) gosper((4*k-3)*factorial(2*k-2)/factorial(k-1),k); 2*(2*k - 1)*factorial(2*(k - 1)) ---------------------------------- factorial(k - 1) gosper(gamma(k+n+2)*n/((k+n+1)*gamma(k+2)*gamma(n+1)),k); gamma(n + 2 + k) --------------------------- gamma(k + 2)*gamma(n + 1) gosper((k+n)*factorial(k+n),k); (n + 1 + k)*factorial(k + n) gosper((3*(1+2*k))/((1+k^2)*(2+2*k+k^2)),k); 3*(k + 2)*k ------------------ 2 2*(k + 2*k + 2) % 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); k 3*3 ------------------- (2*k + 3)*(k + 1) % 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); k 3*3 ------------------------- (2*(k + 1) + a)*(k + 1) gosper(2^k*(k^2-2*k-1)/(k^2*(k+1)^2),k); k 2*2 ---------- 2 (k + 1) gosper((3*k^2+3*k+1)/(k^3*(k+1)^3),k); 2 (k + 3*k + 3)*k ------------------ 3 (k + 1) gosper((6*k+3)/(4*k^4+8*k^3+8*k^2+4*k+3),k); (k + 2)*k ---------------- 2 2*k + 4*k + 3 gosper(-(k^2+3*k+3)/(k^4+2*k^3-3*k^2-4*k+2),k); (2*k + 5)*k -------------- 2 k + 2*k - 1 gosper(k^2*4^k/((k+1)*(k+2)),k); k 4*4 *(k - 1) -------------- 3*(k + 2) gosper((2*k-1)^3,k); 2 2 (2*k - 1)*k gosper(3*k^2+3*k+1,k); 2 (k + 3*k + 3)*k gosper((k^2+4*k+2)*2^k,k); k 2 2*2 *(k + 1) % gosper(2^k*(k^3-3*k^2-3*k-1)/(k^3*(k+1)^3),k); gosper(k*n^k,k); k n *((n - 1)*k - 1)*n ---------------------- 2 (n - 1) % 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); 2 - (k - 2) ------------- k gosper(1/(k*(k+1)),k); k ------- k + 1 gosper(1/(k*(k+2)),k); (3*k + 5)*k ------------------- 4*(k + 2)*(k + 1) gosper(1/(k*(k+10)),k); 9 8 7 6 5 4 ((7381*k + 380755*k + 8495520*k + 107353950*k + 844356513*k + 4272540195*k 3 2 + 13854467330*k + 27627865100*k + 30558324456*k + 14171968800)*k)/(25200 *(k + 10)*(k + 9)*(k + 8)*(k + 7)*(k + 6)*(k + 5)*(k + 4)*(k + 3)*(k + 2) *(k + 1)) % gosper(1/(k*(k+30)),k); gosper(1/(k*(k+1)*(k+2)),k); (k + 3)*k ------------------- 4*(k + 2)*(k + 1) gosper(1/(k*(k+1)*(k+2)*(k+3)*(k+4)*(k+5)*(k+6)*(k+7)*(k+8)*(k+9)* (k+10)),k); 8 7 6 5 4 3 2 ((k + 44*k + 836*k + 8954*k + 59279*k + 249986*k + 667084*k + 1071576*k + 966240)*(k + 11)*k)/(36288000*(k + 10)*(k + 9)*(k + 8)*(k + 7)*(k + 6) *(k + 5)*(k + 4)*(k + 3)*(k + 2)*(k + 1)) gosper(pochhammer(k-n,n),k); pochhammer(k - n,n)*k ----------------------- n + 1 gosper((a+k-1)*pochhammer(a,k),k); (a + k)*pochhammer(a,k) gosper((a-k-1)/pochhammer(a-k,k),k); - 1 --------------------- pochhammer(a - k,k) gosper(binomial(k,n),k); (k + 1)*binomial(k,n) ----------------------- n + 1 gosper(k*binomial(k,n),k); (k*n + k + n)*(k + 1)*binomial(k,n) ------------------------------------- (n + 2)*(n + 1) % gosper(k^10*binomial(k,n),k); gosper(1/binomial(k,n),k); n - 1 - k ----------------------- (n - 1)*binomial(k,n) gosper(k/binomial(k,n),k); (n - 1 - k)*k ----------------------- (n - 2)*binomial(k,n) % gosper(k^10/binomial(k,n),k); gosper(binomial(k-n,k),k); (n - 1 - k)*binomial(k - n,k) ------------------------------- n - 1 gosper((-1)^k*binomial(n,k),k); k - ( - 1) *(k - n)*binomial(n,k) ---------------------------------- n gosper((-1)^k/binomial(n,k),k); k ( - 1) *(k + 1) ----------------------- (n + 2)*binomial(n,k) gosper((-1)^(k+1)*(4*k+1)*factorial(2*k)/ (factorial(k)*4^k*(2*k-1)*factorial(k+1)),k); k - ( - 1) *factorial(2*k) ---------------------------------- k 4 *factorial(k + 1)*factorial(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); (n + 1 - k)*(binomial(n + 1,k) - 2*binomial(n,k)) --------------------------------------------------- n 2*2 *(n + 1 - 2*k) % Vandermonde summand:=binomial(n,k)^2/binomial(2*n,n)$ gosper(sub(n=n+1,summand)-summand,k); 2 2 ((binomial(n + 1,k) *binomial(2*n,n) - binomial(2*(n + 1),n + 1)*binomial(n,k) ) 2 *(2*k - 3*n - 1)*(k - n - 1) )/( 2 (2*(2*(n + 1) - k)*(2*n + 1)*k - (3*n + 1)*(n + 1) ) *binomial(2*(n + 1),n + 1)*binomial(2*n,n)) % 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); ff(k) --------------- (a - e)*gg(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); ff(k) --------------- (a - e)*gg(k) gosper(ff(k-1)/gg(k+1),k); ***** Gosper algorithm: no closed form solution exists % 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(k) --------------- (a - f)*gg(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); ff(k) ----------------------- (2*c + 7 - 4*b)*gg(k) clear(ff,gg); % 4) Examples for which gosper decides that no hypergeometric term % antidifference exists gosper(factorial(k),k); ***** Gosper algorithm: no closed form solution exists gosper(factorial(2*k)/(factorial(k)*factorial(k+1)),k); ***** Gosper algorithm: no closed form solution exists % gosper(1/(factorial(k)*(k^4+k^2+1)),k); gosper(binomial(A,k),k); ***** Gosper algorithm: no closed form solution exists gosper(1/k,k); ***** Gosper algorithm: no closed form solution exists gosper((1+k)/(1-k),k); ***** Gosper algorithm: no closed form solution exists % 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 algorithm: no closed form solution exists gosper(1/(k*(k+1/2)),k); ***** Gosper algorithm: no closed form solution exists gosper(pochhammer(a,k),k); ***** Gosper algorithm: no closed form solution exists gosper(binomial(n,k),k); ***** Gosper algorithm: no closed form solution exists % 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); 2*summ(n - 1) - summ(n) sumrecursion(k*binomial(n,k),k,n); recursion valid for n>=2 (n - 1)*summ(n) - 2*summ(n - 1)*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); 2*(2*n - 1)*summ(n - 1) - summ(n)*n sumrecursion(binomial(n,k)^2/binomial(2*n,n),k,n); summ(n - 1) - summ(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); - ((n - 1 - c)*(c - n)*summ(n) + summ(n - 2)) + (n - 1 - b)*(n - 1 - c)*summ(n - 1) sumrecursion( pochhammer(a,k)*pochhammer(b,k)/(factorial(k)*pochhammer(c,k)),k,a); (b - c + a)*summ(a) - (a - c)*summ(a - 1) % 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); summ(n - 1) - summ(n) sumrecursion((-1)^k/( factorial(2*n+c-1-k)*factorial(2*n-k)*factorial(c+k-1)* factorial(k)),k,n); (2*n - 1 + c)*(2*(n - 1) + c)*(n - 1 + c)*summ(n)*n + summ(n - 1) % 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); - (n - 1 + c - a)*(n - 1 + c - b)*summ(n - 1) + (n - 1 + c)*summ(n)*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); (c + n + b)*summ(n - 1) - summ(n)*n sumrecursion((-1)^(k-n)*binomial(2*n,k)^3,k,n); 2 3*(3*n - 1)*(3*n - 2)*summ(n - 1) - summ(n)*n sumrecursion( (-1)^(k-n)*binomial(2*n,k)^3/(binomial(3*n,n)*binomial(2*n,n)),k,n); summ(n - 1) - summ(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); 3 3 - ((n - 1) *summ(n - 2) + summ(n)*n ) 2 + (17*n - 17*n + 5)*(2*n - 1)*summ(n - 1) % 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); summ(n - 1) + summ(n) sumrecursion((-1)^k*binomial(n,k)*binomial(2*k,n),k,n); 2*summ(n - 1) + summ(n) sumrecursion((-1)^k*binomial(n,k)*binomial(k,j)^2,k,n); 2 (n - 1 - 2*j)*summ(n - 1)*n - (j - n) *summ(n) sumrecursion(binomial(n,k)*binomial(a,k),k,n); (a + n)*summ(n - 1) - summ(n)*n sumrecursion((3*k-2*n)*binomial(n,k)^2*binomial(2*k,k),k,n); summ(n - 1) - summ(n) sumrecursion(binomial(n-k,k),k,n); summ(n - 1) - summ(n) + summ(n - 2) sumrecursion(binomial(n,k)*binomial(n+k,k),k,n); 6*summ(n - 1)*n - 3*summ(n - 1) - summ(n)*n - (n - 1)*summ(n - 2) % 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); summ(n - 1) - summ(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); 5*summ(n - 1) - summ(n) - 4*summ(n - 2) sumrecursion(binomial(n,k)*binomial(2*k,k)*(-1/4)^k,k,n); (2*n - 1)*summ(n - 1) - 2*summ(n)*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); 2*(2*n - 1)*summ(n - 1) - summ(n)*n sumrecursion((-1)^k*binomial(n,k)/binomial(k+a,k),k,n); summ(n - 1) + summ(n) % sumrecursion((-1)^k*binomial(n,k)/binomial(k+a,k),k,a); sumrecursion((-1)^(n-k)*binomial(2*n,k)^2,k,n); 2*(2*n - 1)*summ(n - 1) - summ(n)*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); - (n + 1 + c - a)*(b - c + a)*summ(a) + (a - c)*(a - 1)*summ(a - 1) % 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); (2*p + 1 - 2*n)*(n - p)*summ(n) - 2*(p + 1 - 2*n)*(2*n - p)*summ(n - 1) % 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); (s - t + r)*summ(r) - (r + s)*summ(r - 1) % 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); (2*n - 3 - 2*m)*(n - 1 - m)*summ(n - 1) - (2*n - 1)*summ(n)*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); (j - n)*summ(n) + 2*summ(n - 1)*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); 2 2 2 - ((x + 1) *(x - 1) *summ(n - 2) + 16*summ(n)) + 8*(x + 1)*summ(n - 1) sumrecursion(binomial(n+k-1,2*k-1)*(x-1)^(2*k)*x^(n-k)/k,k,n); 2 2 - ((n - 2)*summ(n - 2)*x + summ(n)*n) + (n - 1)*(x + 1)*summ(n - 1) sumrecursion( 1/(k+1)*binomial(2*k,k)/(n-k+1)*binomial(2*n-2*k,n-k),k,n); summ(n - 1) + summ(n) sumrecursion(binomial(m,r)*binomial(n-r,n-r-q)*(t-1)^r,r,m); (t + 1 + n*t - (t - 1)*q - (t + 1)*m)*summ(m - 1) - ((n + 1 - m)*summ(m) - (m - 1)*summ(m - 2)*t) % 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); (n - 1 + 2*b)*summ(n) - 2*(n - 1 + b)*summ(n - 1) % 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); 2 3*(3*n - 2)*(3*n - 4)*summ(n - 2) + summ(n)*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); - (n + 1 - l - c - b - a)*(n + 2 - 2*l - 2*c - 2*b)*summ(c - 1) + 2*(n + 1 - 2*l - c - 2*b - a)*(n + 1 - c - b - a)*summ(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); (b - c + a)*summ(a) - (a - c)*summ(a - 1) % 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); recursion valid for n>=3 (b + n)*summ(n) - summ(n - 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); k (k + 7)*factorial(---) 7 extended_gosper(k*factorial(k/2),k,2); k (k + 2)*factorial(---) 2 extended_gosper(k*factorial(k/2),k); k k - 1 (k + 2)*factorial(---) + (k + 1)*factorial(-------) 2 2 extended_gosper(binomial(k/2,n),k); k k - 1 (k + 2)*binomial(---,n) + (k + 1)*binomial(-------,n) 2 2 ------------------------------------------------------- 2*(n + 1) extended_gosper(binomial(n,k/2)-binomial(n,k/2-1),k); k - 3 k - 1 ( - ((2*n + 3 - k)*(n + 1 - k)*(binomial(n,-------) - binomial(n,-------)) 2 2 k - 2 k + (n + 2 - k)*(2*(n + 1) - k)*(binomial(n,-------) - binomial(n,---))))/(2 2 2 *(n + 2 - k)*(n + 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); recursion valid for n>=3 summ(n - 1) + 2*summ(n) extended_sumrecursion(binomial(n/2,k),k,n,2,1); 2*summ(n - 2) - summ(n) sumrecursion(binomial(n/2,k),k,n); 2*summ(n - 2) - summ(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); pochhammer( - b + c,n) ------------------------ pochhammer(c,n) % Saalschuetz hypersum({a,b,-n},{c,1+a+b-c-n},1,n); pochhammer( - a + c,n)*pochhammer( - b + c,n) ----------------------------------------------- pochhammer( - a - b + c,n)*pochhammer(c,n) % Kummer hypersum({a,-n},{1+a+n},-1,n); pochhammer(a + 1,n) ----------------------- a + 2 pochhammer(-------,n) 2 % Dixon hypersum({a,b,-n},{1+a-b,1+a+n},1,n); a - 2*b + 2 pochhammer(a + 1,n)*pochhammer(-------------,n) 2 ------------------------------------------------- a + 2 pochhammer(a - b + 1,n)*pochhammer(-------,n) 2 % 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); pochhammer(a - c - d + 1,n)*pochhammer(a + 1,n) ------------------------------------------------- pochhammer(a - c + 1,n)*pochhammer(a - d + 1,n) hypersum({a,1+a/2,d,-n},{a/2,1+a-d,1+a+n},-1,n); pochhammer(a + 1,n) ------------------------- pochhammer(a - d + 1,n) % m-fold case: hypersum({-n,-n,-n},{1,1},1,n); n/2 2 n 1 n ( - 27) *pochhammer(---,---)*pochhammer(---,---) 3 2 3 2 {----------------------------------------------------, n 2 factorial(---) 2 0} % 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); hypergeom({ - n, - n, - n},{1,1},-1) sumtohyper(binomial(n,k)/2^n-sub(n=n-1,binomial(n,k)/2^n),k); - n + 2 - n - hypergeom({----------, - n},{------},-1) 2 2 --------------------------------------------- n 2 sumtohyper(binomial(k+j-1,k-j)*2*(-1)^(j+1)*factorial(2*j-1)/ factorial(j-1)/factorial(j+1)*x^j,j); hypergeom({k + 1, - k + 1},{3},x)*k*x % 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; 4: 4: 4: 4: 4: 4: 4: 4: 4: Time for test: 153560 ms, plus GC time: 7410 ms 5: 5: Quitting Sat Apr 10 01:36:21 MET DST 1999