Tue Apr 15 00:34:42 2008 run on win32
% Test file for the REDUCE package QSUM
%
% Copyright (c) Wolfram Koepf, Harald Boeing, Konrad-Zuse-Zentrum Berlin, 1997
%
% Implementation of the q-Gosper and q-Zeilberger algorithms
%
% Reference:
%
% Koornwinder, T. H.:
% On Zeilberger's algorithm and its q-analogue: a rigorous description.
% J. of Comput. and Appl. Math. 48, 1993, 91-111.
%
% Some examples are from
%
% Koekoek, R. and Swarttouw, R.F.:
% The Askey-scheme of Hypergeometric Orthogonal Polynomials and its q-analogue.
% Report 94-05, Technische Universiteit Delft, Faculty of Technical Mathematics
% and Informatics, Delft, 1994.
%
% Gasper, G. and Rahman, M.:
% Basic Hypergeometric Series.
% Encyclopedia of Mathematics and its Applications 35.
% Ed. by G.-C. Rota, Cambridge University Press, London and New York, 1990.
% Results of manual qsum.tex
%
load qsum;
qgosper(qpochhammer(a,q,k)*q^k/qpochhammer(q,q,k),q,k);
k
(q *a - 1)*qpochhammer(a,q,k)
-------------------------------
(a - 1)*qpochhammer(q,q,k)
qgosper(qpochhammer(a,q,k)*qpochhammer(a*q^2,q^2,k)*
qpochhammer(q^(-n),q,k)*q^(n*k)/(qpochhammer(a,q^2,k)*
qpochhammer(a*q^(n+1),q,k)*qpochhammer(q,q,k)),q,k);
k*n k k n 1 2 2
( - q *(q *a - 1)*(q - q )*qpochhammer(----,q,k)*qpochhammer(a*q ,q ,k)
n
q
2*k n n
*qpochhammer(a,q,k))/((q *a - 1)*(q - 1)*qpochhammer(q *a*q,q,k)
2
*qpochhammer(a,q ,k)*qpochhammer(q,q,k))
qgosper(qpochhammer(q^(-n),q,k)*z^k/qpochhammer(q,q,k)*z^n,q,k);
***** No q-hypergeometric antidifference exists.
off qgosper_down;
qgosper(q^k*qbrackets(k,q),q,k);
k k
- q *(q + 1 - q )*qbrackets(k,q)
-----------------------------------
k
(q - 1)*(q + 1)*(q - 1)
on qgosper_down;
qgosper(q^k,q,k,0,n);
n
q *q - 1
----------
q - 1
qsumrecursion(qpochhammer(q^(-n),q,k)*z^k/qpochhammer(q,q,k),q,k,n);
n n
- ((q - z)*summ(n - 1) - q *summ(n))
on qsumrecursion_certificate;
proof:=qsumrecursion(qpochhammer(q^(-n),q,k)*z^k/qpochhammer(q,q,k),q,k,n);
n n
proof := { - ((q - z)*summ(n - 1) - q *summ(n)),
k n
- (q - q )*z
----------------,
n
q - 1
k 1
z *qpochhammer(----,q,k)
n
q
--------------------------,
qpochhammer(q,q,k)
k,
downward_antidifference}
off qsumrecursion_certificate;
% proof of statement
lhside:= qsimpcomb(sub(summ(n)=part(proof,3),
summ(n-1)=sub(n=n-1,part(proof,3)),part(proof,1)));
k k n n 1
z *(q *(q - z) + q *(z - 1))*qpochhammer(----,q,k)
n
q
lhside := -----------------------------------------------------
n
(q - 1)*qpochhammer(q,q,k)
rhside:= qsimpcomb((part(proof,2)*part(proof,3)-
sub(k=k-1,part(proof,2)*part(proof,3))));
k k n n k 1
- z *((q - q )*z - q *(q - 1))*qpochhammer(----,q,k)
n
q
rhside := ---------------------------------------------------------
n
(q - 1)*qpochhammer(q,q,k)
qsimpcomb((rhside-lhside)/part(proof,3));
0
% proof done
operator qlaguerre, qcharlier;
% q-Laguerre polynomials, Koekoek, Swarttouw (3.21)
qsumrecursion(qpochhammer(q^(alpha+1),q,n)/qpochhammer(q,q,n),
{q^(-n)}, {q^(alpha+1)}, q, -x*q^(n+alpha+1), qlaguerre(n));
n alpha + n n
((q + 1 - q )*q - q *(q *x + q))*qlaguerre(n - 1)
alpha + n n
+ ((q - q)*qlaguerre(n - 2) + (q - 1)*qlaguerre(n))*q
% q-Charlier polynomials, Koekoek, Swarttouw (3.23)
qsumrecursion({q^(-n),q^(-x)},{0},q,-q^(n+1)/a,qcharlier(n));
x n n 2*n
- ((q *((q + 1 - q )*a + q )*q - q )*qcharlier(n - 1)
x n n
+ q *((q + a*q)*(q - q)*qcharlier(n - 2) - qcharlier(n)*a*q))
% continuous q-Jacobi polynomials, Koekoek, Swarttouw (3.10)
%% on qsum_nullspace;
%% term:= qpochhammer(q^(alpha+1),q,n)/qpochhammer(q,q,n)*
%% qphihyperterm({q^(-n),q^(n+alpha+beta+1),
%% q^(alpha/2+1/4)*exp(I*theta), q^(alpha/2+1/4)*exp(-I*theta)},
%% {q^(alpha+1), -q^((alpha+beta+1)/2), -q^((alpha+beta+2)/2)},
%% q, q, k)$
%% qsumrecursion(term,q,k,n,2);
%% off qsum_nullspace;
% Some more qgosper results with proof
%
% Gasper, Rahman (2.3.4)
term:=qpochhammer(a,q,k)*qpochhammer(a*q^2,q^2,k)*qpochhammer(q^(-n),q,k)*
q^(n*k)/(qpochhammer(a,q^2,k)*qpochhammer(a*q^(n+1),q,k)*qpochhammer(q,q,k));
k*n 1 2 2
q *qpochhammer(----,q,k)*qpochhammer(a*q ,q ,k)*qpochhammer(a,q,k)
n
q
term := ----------------------------------------------------------------------
n 2
qpochhammer(q *a*q,q,k)*qpochhammer(a,q ,k)*qpochhammer(q,q,k)
result:=qgosper(qpochhammer(a,q,k)*qpochhammer(a*q^2,q^2,k)*
qpochhammer(q^(-n),q,k)*q^(n*k)/
(qpochhammer(a,q^2,k)*qpochhammer(a*q^(n+1),q,k)*qpochhammer(q,q,k)),q,k);
k*n k k n 1
result := ( - q *(q *a - 1)*(q - q )*qpochhammer(----,q,k)
n
q
2 2 2*k n
*qpochhammer(a*q ,q ,k)*qpochhammer(a,q,k))/((q *a - 1)*(q - 1)
n 2
*qpochhammer(q *a*q,q,k)*qpochhammer(a,q ,k)*qpochhammer(q,q,k))
qsimpcomb(result-sub(k=k-1,result)-term);
0
% Gasper, Rahman (3.8.16)
term:=(1-a*c*q^(4*k))*(1-b/c*q^(-2*k))*qpochhammer(a,q,k)*qpochhammer(b,q,k)*
qpochhammer(q^(-3*n),q^3,k)*qpochhammer(a*c^2/b*q^(3*n),q^3,k)*q^(3*k)/
((1-a*c)*(1-b/c)*qpochhammer(c*q^3,q^3,k)*qpochhammer(a*c/b*q^3,q^3,k)*
qpochhammer(a*c*q^(3*n+1),q,k)*qpochhammer(b/c*q^(1-3*n),q,k));
3*n 2
k q *a*c 3 1 3
term := (q *qpochhammer(-----------,q ,k)*qpochhammer(------,q ,k)
b 3*n
q
*qpochhammer(a,q,k)*qpochhammer(b,q,k)
3
6*k 2 4*k 2*k a*c*q 3
*( - q *a*c + q *a*b*c + q *c - b))/(qpochhammer(--------,q ,k)
b
b*q 3*n
*qpochhammer(--------,q,k)*qpochhammer(q *a*c*q,q,k)
3*n
q *c
3 3 2
*qpochhammer(c*q ,q ,k)*(a*b*c - a*c - b + c))
result:=qgosper((1-a*c*q^(4*k))*(1-b/c*q^(-2*k))*qpochhammer(a,q,k)*
qpochhammer(b,q,k)*qpochhammer(q^(-3*n),q^3,k)*qpochhammer(a*c^2/
b*q^(3*n),q^3,k)*q^(3*k)/((1-a*c)*(1-b/c)*qpochhammer(c*q^3,q^3,k)*
qpochhammer(a*c/b*q^3,q^3,k)*qpochhammer(a*c*q^(3*n+1),q,k)*
qpochhammer(b/c*q^(1-3*n),q,k)),q,k);
3*k + 3*n 2 2*k k + n 2*n k k
result := ( - (q *a*c - b)*(q + q + q )*(q *a - 1)*(q *b - 1)
3*n 2
k n q *a*c 3 1 3
*(q - q )*qpochhammer(-----------,q ,k)*qpochhammer(------,q ,k)
b 3*n
q
3*n
*qpochhammer(a,q,k)*qpochhammer(b,q,k)*c)/((q *a*c - b)
3
3*n a*c*q 3
*(q *c - 1)*(a*c - 1)*(b - c)*qpochhammer(--------,q ,k)
b
b*q 3*n
*qpochhammer(--------,q,k)*qpochhammer(q *a*c*q,q,k)
3*n
q *c
3 3
*qpochhammer(c*q ,q ,k))
qsimpcomb(result-sub(k=k-1,result)-term);
0
end;
Time for test: 629 ms, plus GC time: 27 ms