Basic hypergeometric series

1. Series[QPochhammer[q, q], {q, 0, 100}]
   Series[\!\(<br> \*UnderoverscriptBox[\(\[Product]\), \(k = 1\), \(100\)]\((1 -<br>     q^k)\)\), {q, 0, 100}]
   f[q_] := \!\(<br> \*UnderoverscriptBox[\(\[Sum]\), \(k = 0\), \(100\)]\(PartitionsP[<br>     k] q^k\)\)
   Series[1/QPochhammer[q, q], {q, 0, 100}]
   Series[f[q], {q, 0, 100}]
   d[n_] := DivisorSigma[1, n]
   g[q_] := \!\(<br> \*UnderoverscriptBox[\(\[Sum]\), \(k = 1\), \(100\)]\(d[k] q^k\)\)
   Expand[f[q]*g[q]]
2.  

 

q-hypergeometric series

\(\sum_{n\geq 0}^{\infty}\frac{q^{n^2/2}}{(q)_n}\sim \exp(\frac{\pi^2}{12t}-\frac{t}{48})\)

 

1. f[q_] := QHypergeometricPFQ[{}, {}, q, -q^(1/2)]
   g[q_] := Exp[-(Pi^2/(12 Log[q])) + Log[q]/48]
   Table[N[f[1 - 1/10^(i)]/g[1 - 1/10^(i)], 50], {i, 1, 5}] // TableForm

 

related items

• asymptotic analysis of basic hypergeometric series