Certain Representations of Mock-Theta Functions

Abstract

Recently, Denis [4], making use of the identity

$${e_q}\left( {xt - xyt} \right){e_q}\left( {t - tx} \right) = {e_q}\left( {t - xyt} \right)$$

((1))

where ${e_q}\left( x \right) = \sum\nolimits_{n = 0}^\infty {\frac{{{x^n}}}{{{{[q]}_n}}}}$ with [q]n = (1 − q)(1 − q²)…(1 − qⁿ), for n ≥ 1, [q]₀ = 1, established the following result:

$$\begin{gathered}\sum\limits_{m = 0}^n {\left[ {\begin{array}{*{20}{c}}n \\m \\\end{array} } \right]} \frac{{{x^m}{{[(a)]}_m}}}{{{{[(b)]}_m}}}A + 1{\Phi _B}\left[ {\begin{array}{*{20}{c}}{{q^{ - n + m}},} & {(a){q^m};x{q^{n - m}}} \\{} & {(b){q^m}} \\\end{array} } \right] \hfill \\{\times _{C + 1}}{\Phi _D}\left[ {\begin{array}{*{20}{c}}{{q^{ - m}},} & {(c);y{q^m}} \\{} & {(d)} \\\end{array} } \right] \hfill \\{ = _{A + C + 1}}{\Phi _{B + D}}\left[ {\begin{array}{*{20}{c}}{{q^{ - n}},} & {(a),(c);xy{q^n}} \\{} & {(b),(d)} \\\end{array} } \right]. \hfill \\\end{gathered}$$

((2))

with A = B, C = D and $\left[ {\begin{array}{*{20}{c}}n \\m \\\end{array} } \right]$ is the q-binomial coefficient, defined by [q] _n /[q] _m [q]_n−m and the ϕ-functions are the usual basic hypergeometric functions (cf. Section 2 for detailed definitions). The parameters of the type (a _m ), m ∈ N in small brackets shall stand for the sequence of m parameters a₁, a₂, …, a _m . If m = A, we shall denote it by (a) instead of (a _A ).