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\(<< c:\marko\gosper\newaisb.m\)

The q-factorial

FS[qFac[q, q, n + 1] - qFac[q, q, n]]

GosperSum[%, n, {q}]

An example which is both hypergeometric and q-hypergeometric

GosperSum[(q^n), n]

GosperSum[(q^n), n, {q}]

Axel Riese's first bibasic example

GosperSum[
      (1 - a p^k q^k) 
        (1 - b p^k /q^k)/(1 - a)/(1 - b) qFac[a, p, k] 
        qFac[b, p, k] qFac[c, q, k] 
        qFac[a/(b c), q, k]/qFac[q, q, k]/qFac[a q/b, q, k]/
            qFac[a p/c, p, k]/qFac[b c p, p, k] q^k, {k, 0, n}, {p, q}]

Bill Gosper's tetrabasic example

f[k_] := 
      b^k q^Binomial[k + 1, 2] (1 - a p^k/b/q^k) 
        (1 - a b p^k q^k) (1 - d s^k/c /r^k) 
        (1 - c d r^k s^k) qFac[a c, p r, k] qFac[a/c, p/r, k] 
        qFac[d b, s q, k] 
        qFac[d/b, s/q, k]/
          (a^k p^Binomial[k + 1, 2] qFac[b c q r, q r, k] 
              qFac[b/c q/r, q/r, k] qFac[d a s p, s p, k] 
              qFac[d/a s/p, s/p, k])

GosperSum[f[k], {k, 0, n}, {p, q, r, s}]

Table[Together[Sum[f[k], {k, 0, n}]] - %,  {n, 0, 2}];

Together /@ %

%%% // Simplify

Bill Gosper's tribasic example

f[k_] := 
      p^Binomial[k, 2] (1 - a b p^k q^k) (1 - a c p^k r^k) 
        (-a)^k qFac[b, q, k] 
        qFac[c, r, k]/qFac[a b c p q r, p q r, k]/
          qFac[a p, p, k]

GosperSum[f[k], k, {p, q, r}]

FS[(% /. k -> k + 1) - %  - f[k]]

PowerExpand[%]

GosperSum[f[k], {k, 0, n}, {p, q, r}]

% // Simplify

Table[Together[Sum[f[k], {k, 0, n}] - %], {n, 0, 3}]