Unified Theory of Recurrence Formulas. I

Teturo Inui
1950 Progress of theoretical physics  
Introduction and Sumniary. Among several transcendental functions appearing in mathematical physics, those of Sturm-Liouville type may be {:alled. most distinguished. since many of the important properties such as .. orthogonality relation ", "normalization integral", .. asymptotic expansion II etc. are deduced in unified way from general standpoint, not relying upon special form of individual function. Now, such is lacked, in the literature, concerning the deduction of the important property:
more » ... Recurrence Formulas" RF. It is the purpose ofthe present paper to demonstrate that there exists a unified method. The idea of the writer was stimulated by reading interesting papers of E. Schrodinger and L. Infeld. E. Sc'hrodinger(1) first proposed "a new method of determining quantum mechanical eigen-value problem". By his method, a series of quantum mechanical eigen-functions were solved without applying traditional series method. The fundamental method of this author is the factorization of the principal part of given linear differential equation of the second order. Each factor has important property, fot which reason will be denoted by a name: .. stair operator" SO. The investigation by Infeld(2) is a systematic study of a certain type of factorization. In a paper presented to Physico-mathematical Society of Japan, the writer(~) has generalized the tnethod of Infeld, so that the results are applicable to all transcendental functions of hypergeometric and confluent hypergeometric types. But, what is the reason fo possibility of such factorization? The result shows, from view point of this paper, that key to magic factorization of SchrOdinger-Infeld is in our hands so long as we are concerned to differential equations with three regular singularities or those with one regular singularity and one irregular singularity.
doi:10.1143/ptp.3.168 fatcat:s6i3qz2i3ncz7cfn3s6pwofqy4