q–GAUSS SUMMATION VIA RAMANUJAN AND COMBINATORICS

where |c/(ab)| < 1. Gauss’s name is attached to this theorem, because it is the qanalogue of Gauss’s summation for ordinary or Gaussian hypergeometric series. The theorem (1.1) was however first discovered by E. Heine [8] in 1847. We only know of two proofs of (1.1) up to recent times. The first proof, due to Heine [8], uses what we now call Heine’s transformation, and this proof can be found in the texts of G. E. Andrews [1, p. 20], Andrews, R. Askey, and R. Roy [2, p. 522], and G. Gasper and M. Rahman [7, p. 10]. The second proof employs the q-analogue of Saalschütz’s theorem and can be read in the texts of W. N. Bailey [3, p. 68] and L. J. Slater [10, p. 97]. The purpose of this short note is to present two different approaches to the proof of the q-Gauss summation theorem. The first proof is due to Ramanujan and is published in a fragment with his lost notebook [9, pp. 268–269]. Ramanujan’s proof, which encompasses Corollary ??, Lemma 2.3, and Theorem 2.4 in Section 2, does not appear to have been noticed by many mathematicians. The second approach uses the theory of partitions and rests upon a combinatorial interpretation of each side of (1.1). Combinatorial proofs of (1.1) have been given by S. Corteel and J. Lovejoy [6], Corteel [5], and the second author [11]. The new concept of an overpartition was employed in the latter three papers. Here, without the use of overpartitions, a variation of the second author’s proof [11] is presented.