著者

E. E. Kummer

出版者

THE PHYSICAL SOCIETY OF JAPAN , The Mathematical Society of Japan

雑誌

Tokyo Sugaku-Butsurigaku Kwai Kiji (ISSN:21852669)

巻号頁・発行日

vol.4, no.6, pp.353-405, 1891 (Released:2006-03-27)

著者

E. E. Kummer

出版者

THE PHYSICAL SOCIETY OF JAPAN , The Mathematical Society of Japan

雑誌

Tokyo Sugaku-Butsurigaku Kwai Kiji (ISSN:21852669)

巻号頁・発行日

vol.4, no.5, pp.276-325, 1890 (Released:2006-03-27)

---

The hypergeometric series, which forms the subject of this paper, has for a long time attracted the attention of mathematicians, comprising, as it does, a very large number of series of well known functions, and furnishing, in addition, the means of integrating a special class of liner differential equations of the second order. Euler was the first to give, in Actis Academiae Petrop., two general transformatric series of the same kind. These transformarions were afterward derived and proved in various ways by Pfaff, Jacobi and Gudermann. Moreover Gauss published in Comment. Soc. Gotting. (Tome II. a. 1812.) his investigations on this series. It is well known that this paper contains, in a brief space, not only a large number of the fundamental properties of the hypergeometric series, but also the thoroughly complete theories of two transcendental functions closely allied with it, and it's application to continuted fractions and definite integrals. This paper is however only the first installment of a greater memoir which has not as yet been published. Especially it lacks the comparison of such hypergeometric series with each other, in which the last element x is different. This will consequently be the chief object of the present paper. The numerous applications of the formula which we shall derive, are chiefly concerned with elliptic transcendents, a large number of which is contained in the general hypergeometric series. The results obtained by Gauss in the above-mentioned paper will, wherever it is necessary, be assumed as known. We shall also retain the notations introduced by Gauss.