摘要:

In this article, by taking the famous two Roger—Ramanujan identities as an example, the author considered which type of functions admit an infinite production expression in the sense ofq‐analogue theory. The key observation is that the seriescan be expressed as the determinant of an infinite matrix whose elements decrease geometrically in diagonal direction. In casesa= 1 anda=q, they coincide with the twoR‐Ridentities. As a result, it is shown that the series has already an infinite product expression. The proof is rather elementary. In fact, first, we decompose the given matrixA(a,q) asPis an infinite matrix having a decreasing geometric sequence on the semidiagonal away from the main diagonal by one row under.Qis a similar matrix but above.Secondly, we interchange determinant and trace through exponential map. Then we need to calculate the trace of matrices which are products ofPandQ. In short, we attribute the problem to calculate the sums of geometric sequences. The remaining question is to investigate I.T.[B1B2⋯B2k] whereBi= PorQin a calculable style. For the meaning of I.T., refer to the infinite product expression which the author obtained. The merit of this argument exists in the fact that the determinant of any matrix stated in the title is always expressible in an infinite product form. That is to say, a great many functions have an infinite product exp

ISSN:0022-2526    

DOI:10.1002/sapm1988783167

年代:1988

数据来源: WILEY

摘要:

This paper aims at presenting a general class of mixed generating functions for the Jacobi polynomials. It is also shown how the main generating function can be suitably applied to yield numerous further results involving Jacobi polynomials and various other polynomials associated with them.

ISSN:0022-2526    

DOI:10.1002/sapm1988783177

年代:1988

数据来源: WILEY

摘要:

This paper describes techniques that can be used to transform PDEs with variable coefficients into equations with constant coefficients. The techniques are illustrated by calculating shear flows over quite general surfaces, by solving the signaling problem for diffusive processes in inhomogeneous materials, and by solving the signaling problem for acoustical waves when the sound speed varies with distance. The techniques may also be used to solve equations governing processes in inhomogeneous, anisotropic materials.

ISSN:0022-2526    

DOI:10.1002/sapm1988783183

年代:1988

数据来源: WILEY

摘要:

The generalized wave equations and generalized sine‐Gordon equations are known to be associated to linear spectral problems. In this paper we show that the geometric Backlund transformation for the nonlinear equations corresponds to changing the discrete part of the scattering data for the linear problem

ISSN:0022-2526    

DOI:10.1002/sapm1988783227

年代:1988

数据来源: WILEY