Some basic hypergeometric identities needed in "Another refinement of the Bender-Knuth (ex-)Conjecture"

<< hyp.q;

Definitions

w[n_, c_, k_] := 
      pq[q^(k + 1), n - 1]*
        pq[q^(k - c - n + 1), 
            n - 1]/(1 - q)^(2  n - 2)

x[n_, c_, k_] := 
      SUM[q^Binomial[i, 2]*pq[q^(k + 1), n - 1]*
          pq[q^(k - c - i + 1), 
            i - 1]*pq[q^(k - c - n + 1), n - i - 1]/pq[q, i - 1]/
              pq[q, n - 1 - i]/pq[q^(c + i + 1), n - 1], {i, 1, 
          Infinity}]

y[n_, c_, k_] := 
      SUM[q^Binomial[i, 2]*pq[q^(k + 1), i - 1]*
          pq[q^(k + i + 1), 
            n - i - 1]*pq[q^(k - c - n + 1), n - 1]/pq[q, i - 1]/
              pq[q, n - 1 - i]/pq[q^(c + i + 1), n - 1], {i, 1, 
          Infinity}]

u[n_, c_, 
        k_] := (-1)^n*
          q^((n - 1)*(2*c + n)/2)*((-1)^c*x[n, c, k] - 
              y[n, c, k]) + (-1)^k*q^((n - 1)*k)

Identity (7.1)

SUM[w[n, c, k]*q^k, {k, 0, Infinity}]

%  /.  SUMph

%  /.  S2101

Identity (7.2)

x[n, c, 0]

%  /.  SUMph

%  /.  C34[1, 1]

%  /.  S2104

Expand[%]  /.  zus1    /.  zus2   /.  zus3

y[n, c, 0]

%  /.  SUMph

%  /.  C02

% /.  C15[2]

%  /.  S2104

Identity (7.3)

x[n, c, -n]

%  /.  SUMph

%  /.  C34[1, 1]

%  /.  S2104

y[n, c, -n]

%  /.  SUMph

%  /.  C15[2]

%  /.  S2104

Identity (7.4)

x[n, c, 1]

%  /.  SUMph

%  /.  C14[1]

%  /.  C42[1, 1]

%  /.  S2104

%  /.  C34[1, 1]

%  /.  S2104

y[n, c, 1]

%  /.  SUMph

%  /.  C16[1]

PosListe[%, 2]

Ers[%%, C15[2], {{5, 1}}]

%  /.  S2104

%  /.  C02

%  /.  C41[2, 1]

% /. C11

%  /.  S2104

%  /.  C37[2, 1]

%  /.  S2104

Identity (7.5)

x[n, c, -n - 1]

%  /.  SUMph

%  /.  C14[2]

%  /.  C42[1, 1]

%  /.  S2104

%  /.  C34[1, 1]

%  /.  S2104

y[n, c, -n - 1]

%  /.  SUMph

%  /.  C14[2]

PosListe[%, 2]

Ers[%%, C15[2], {{5, 2}}]

%  /.  S2104

%  /.  C36[2, 1]

PosListe[%, 2]

Ers[%%, C11, {{5, 1}}]

%  /.  S2104

PosListe[%, 3]

Ers[%%, C37[2, 1], {{5, 2, 3}}]

%  /.  S2104

%  /.  C15[1]

%  /.  S2104

Identity (7.6)

Identity (7.12)

SUM[(-1)^i*
        q^(Binomial[n - i + 1, 2] + i)*(1 - q)^(2  n - 1)/
              pq[q^(c + i), n]/pq[q, i - 1]/pq[q, n - i], {i, 1, 
        Infinity}]

%  /.  SUMph

%  /.  S2101

Identity (7.13)

SUM[(-1)^
            i*q^Binomial[n + i, 2]/pq[q^(c + i + 1), n, q]/
              pq[q, i]/pq[q, n - 1 - i], {i, 0, 
          Infinity}]*(1 - q)^(2  n - 1)

%  /.  SUMph

%  /.  S2102

Identity (7.14)

SUM[
      q^Binomial[i, 2]*pq[q^(c + 1), i]*
        pq[q^(c + i + n + 1), 
          n - i - 1]*pq[q^(n - i), i]/pq[q^n, n - 1]/pq[q, i]*
        pq[q^(1 + c + 2*i), 1], {i, 0, Infinity}]

%  /.  SUMph

%  /.  T4308

%  /.  T2101

%  /.  S2104

Identity (7.15)

SUM[
      q^Binomial[i, 2]*pq[q^(c + 1), i]*
        pq[q^(c + i + n + 1), 
          n - i - 1]*pq[q^(n - i), i]/pq[q^n, n - 1]/pq[q, i]*
        pq[q^(2 + 2  c + 4*i), 1]/pq[q^(1 + c + 2*i), 1], {i, 0, 
        Infinity}]

%  /.  SUMph

%  /.  T4310

%  /.  T8704

%  /.  phOrdne  /.   phPerm[1, 2, 5, 7, 3, 4, 6, b]  /.  
      S8702