The Mock Theta Functions (2)
The Mock Theta Functions (2)
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DOI：
10.1112/plms/s2-42.1.274
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发表时间：
1937
影响因子：
1.8
通讯作者：
G. N. Watson
中科院分区：
文献类型：
--
作者：
G. N. Watson
Ramanujan's last letter to Hardy was concerned with mock theta functions of orders three, five, and seven. On a previous occasion* I have dealt with the functions of the third order, and I now come to the functions of the fifth order. Ramannjan defined ten such functions, connected by certain relations. I have succeeded in establishing all the relations whose existence was indicated by Ramanujan; but I have failed to construct a complete and exact transformation theory of the functions, on the lines of the transformation theory of functions of the third order, and, in view of the complexity of all the series which are involved, I am becoming somewhat sceptical concerning the existence of an exact transformation theory for functions of the fifth order. The somewhat nebulous results given in § 6 are all that I have succeeded in obtaining in this direction. After dealing with Ramanujan's work on the functions of the fifth order, I use his results to study the properties of a transcendental function f (v>£\Q, q) previously investigated by Lerch; some special cases of this function are readily expressible as the sum of two mock theta functions. Lerch has shown that it is possible to express his function (with q= q) in terms of theta functions combined with two auxiliary functions^(rc, q) and T1 (tc, q), defined by the formulae oo n2n"/y. 4n vu> i~ n\ y 1 x ol> ffJ" t (l? a)(l-3 4)(l-2 4 n)> l'q)~· Journal London Math. Soc, 11 (1936), 55-80. In that paper I quoted the whole of Ramanujan's letter, with the exception of his statements about functions of orders five and seven; these statements had been published previously in his Collected Papers (1927), 354-355. No reference to mock theta functions is to be found elsewhere in Ramanujan's· writings.
Ramanujan's last letter to Hardy was concerned with mock theta functions of orders three, five, and seven. On a previous occasion* I have dealt with the functions of the third order, and I now come to the functions of the fifth order. Ramannjan defined ten such functions, connected by certain relations. I have succeeded in establishing all the relations whose existence was indicated by Ramanujan; but I have failed to construct a complete and exact transformation theory of the functions, on the lines of the transformation theory of functions of the third order, and, in view of the complexity of all the series which are involved, I am becoming somewhat sceptical concerning the existence of an exact transformation theory for functions of the fifth order. The somewhat nebulous results given in § 6 are all that I have succeeded in obtaining in this direction. After dealing with Ramanujan's work on the functions of the fifth order, I use his results to study the properties of a transcendental function f (v>£\Q, q) previously investigated by Lerch; some special cases of this function are readily expressible as the sum of two mock theta functions. Lerch has shown that it is possible to express his function (with q= q) in terms of theta functions combined with two auxiliary functions^(rc, q) and T1 (tc, q), defined by the formulae oo n2n"/y. 4n vu> i~ n\ y 1 x ol> ffJ" t (l? a)(l-3 4)(l-2 4 n)> l'q)~• Journal London Math. Soc, 11 (1936), 55-80. In that paper I quoted the whole of Ramanujan's letter, with the exception of his statements about functions of orders five and seven; these statements had been published previously in his Collected Papers (1927), 354-355. No reference to mock theta functions is to be found elsewhere in Ramanujan's• writings.