Basic Exponential Functions, "Addition" Theorems, q-Fourier Series and Their Extensions
In this talk a review of basic exponential functions and basic trigonometric functions will be given.
There was a great deal of interest in the last decade in q-analogs of important classical formulas. A few q-analogs of exponential and trigonometric functions are known. The old, F. H. Jackson's q-analogs of the exponential function were introduced at the beginning of the last century. They were studied in details and have been found useful in many applications.
Some new q-exponential and q-trigonometric functions have been introduced recently. They appear from different contexts. Ismail and Zhang expanded the corresponding basic exponential function in terms of "q-spherical harmonics'' and later, together with Rahman, they extended this interesting analog of the expansion formula of the plane wave from q-ultraspherical to continuous q-Jacobi polynomials. "Addition'' theorems for the basic trigonometric functions were found by Suslov and then by Ismail and Stanton who also found a few important integrals involving these basic exponential functions. Bustoz and Suslov have established the orthogonality property and introduced the corresponding basic analog of Fourier series. Numerical investigation of these series have been started recently by Gosper and Suslov with the help of the Macsyma computer algebra system.
In this talk we intend to lay a sound foundation for this study. Interesting open problems and further extensions of the theory will be given.
Extensions of Euler's summation formulas and the zeta function: latex, dvi, ps, pdf