1
2/* ------------------------------ */
3/* Gosper's algorithm's Test File */
4
5g1:(-1)^k*k/(4*k^2-1);
6
7g2:1/(4*k^2-1);
8
9g3:x^k;
10
11g4:(-1)^k*a!/(k!*(a-k)!);
12
13g5:k*k!;
14
15g6:(k+1)*k!/(k+1)!;
16
17/* Non-Gosper summable                                        */
18/* It has been checked against the Mathematica package "Zb"   */
19/* by Paule/Schorn/Riese available at www.risc.uni-linz.ac.at */
20g7:1/((a-k)!*k!);
21
22/* --------------------------------- */
23/* Zeiberger's Algorithm's Test File */
24
25f1 : binomial(n,k);
26
27f2 : binomial(n,k)^2;
28
29f3 : binomial(n,k)^3;
30
31f4 : binomial(n,k)^4;
32
33f5 : binomial(n,k)^5;
34
35f6 : binomial(n,k)^6;
36
37f7 : binomial(n,k)^7;
38
39f8 : binomial(n,k)^8;
40
41f9: binomial(n,k)^9;
42
43f10: binomial(n,k)^10;
44
45/* Binomial theorem */
46h1: binomial(n,k)*x^k;
47
48/* Vandermonde identity recurrence */
49h2: binomial(a, k)* binomial(b, n-k);
50
51/* Dixon's identity */
52h3: binomial(n+b, n+k)* binomial(n+c, c+k)*binomial(b+c, b+k)*(-1)^k;
53
54/* Karlsson-Gosper identity 1 */
55h4: binomial(n,k)*(n-1/4)!/(n-k-1/4)!/(2*n+k + 1/4)!*9^(-k);
56
57/* Karlsson-Gosper identity 2 */
58h5: binomial(n,k) * (n-1/4)! / (n-k-1/4)! / (2*n+k+5/4)! * 9^(-k);
59
60/* Wilson polynomials recurrence */
61h6: binomial(n,k) * (n+a+b+c+d-2+k)! * (a+x+k-1)! * (a-x+k-1)! /
62               (n+a+b+c+d-2)! / (a+b+k-1)! / (a+c+k-1)! / (a+d+k-1)! * (-1)^k;
63
64
65/* Wilson polynomials without constants in k (for debugging) */
66h6b : removeBinomial(h6)/(n!)*((n+d+c+b+a-2)!);
67
68/* Laguerre-Orthogonality */
69h7: (k+k2)! * (n1)! * (n2)! / (k)! / (n1-k)! / (k2)! / (n2-k2)! /
70                (k)! / (k2)! * (-1)^k;
71
72/*  Fibonacci-Recurrence */
73h8: (-k+n)! / k! / (-2*k+n)!;
74
75/* Trinomial coefficients */
76h9: n! / k! / (k+m)! / (-2*k-m+n)!;
77
78/*  First special case of Strehl identity (MEMO, Feb 25, 1992) */
79h10: binomial(n, k)^2 * binomial( 2*k, k);
80
81
82/* Second special case of Strehl identity (MEMO, Feb 25, 1992) */
83h11: binomial(n, k)^2 * binomial( 2*k, k+a);
84
85/* Third special case of Strehl identity (MEMO, Feb 25, 1992) */
86h12: binomial(n, k)^2 * binomial( 2*k, k) * binomial(2*k,n-k);
87
88/* Fibonacci recurrence */
89h13: (n+k)! * n! / k!^3 / (n-k)!^2;
90
91/* Debugging artificial examples */
92d1 : (2*n+k-1)!/(4*n+2*k)!;
93
94d2 : (3*n+k-1)!/(6*n+2*k)!;
95
96
97GOSPER_TEST :
98  [[g1,k,n,0],[g2,k,n,0],[g3,k,n,0],
99   [g4,k,n,0],[g5,k,n,0],[g6,k,n,0]];
100
101EASY_TEST :
102[[f1,k,n,1],[f2,k,n,1],[f3,k,n,2],[f4,k,n,2],
103 [h1,k,n,1],[h2,k,n,1],[h8,k,n,2],[h9,k,n,2],
104 [h10,k,n,2],[h13,k,n,2],[d1,k,n,1],[d2,k,n,1]];
105
106HARD_TEST :
107[[h3,k,n,1],[h4,k,n,1],[h5,k,n,1],[h7,k,n1,1],[h11,k,n,3]];
108
109EXTREME_TEST : [[f5,k,n,3], [h6,k,n,2],[h12,k,n,5]];
110
111FULL_TEST : append(GOSPER_TEST,EASY_TEST,
112                   HARD_TEST,EXTREME_TEST);
113
114kill (g1, g2, g3, g4, g5, g6, g7,
115    f1, f2, f3, f4, f5, f6, f7, f8, f9, f10,
116    h1, h2, h3, h4, h5, h6, h6b, h7, h8, h9, h10, h11, h12, h13,
117    d1, d2);
118