On a Q-Extension of ‘Incomplete’ Beta Function

In a recent paper, R.P. Agarwal [1] has indicated the relationship between an ‘incomplete’ q-beta function and the Ramanujan’s function ∑ (a)_nZⁿ/(b)_n. The q-gamma function has been known for long and was studied in details by RH. Jackson [10,11]. The q-beta function occurs in some of the works of Ramanujan and both these q-functions have been studied, recently, by a number of mathematicians, notably, N.J. Fine [8], Richard Askey [2,4,5,6], G.E. Andrews [2,3], W.A. Al-Salam and A. Verma [16]. However, it seems surprising, as remarked by Agarwal [1], that none of the above authors seem to note, as such, the relationship of their results with an ‘incomplete’ q-beta function. Since the ‘incomplete’ beta functions are very useful in statistical analysis and the q-functions have been shown to have varied applications in Physics and Statistics [Exton, 7], it seems worthwhile putting on record some of the analytical properties of the ‘incomplete’ q-beta functions.