Asymptotics of the generalized Gegenbauer functions of fractional degree Abstract   The generalized Gegenbauer functions of fractional degree (GGFFs), denoted by         r        G    xcexbd     (  xcexbb  )      (  x  )      (right GGFFs) and         l      G    xcexbd     (  xcexbb  )      (  x  )      (left GGFFs) with     x  xe2x88x88   (  xe2x88x92  1  ,  1  )   ,         xcexbb  u003e  xe2x88x92  1  xe2x88x95  2     and real     xcexbd  xe2x89xa5  0  ,     are special functions (usually nonpolynomials), which are defined upon the hypergeometric representation of the classical Gegenbauer polynomial by allowing integer degree to be real fractional degree. Remarkably, the GGFFs become indispensable for optimal error estimates of polynomial approximation to singular functions, and have intimate relations with several families of nonstandard basis functions recently introduced for solving fractional differential equations. However, some properties of GGFFs, which are important pieces for the analysis and applications, are unknown or under explored. The purposes of this paper are twofold. The first is to show that for     xcexbb  u003e  xe2x88x92  1  xe2x88x95  2     and     x    cos  xcexb8     with     xcexb8  xe2x88x88   (  0  ,  xcfx80  )   ,              (  sin  xcexb8  )     xcexbb          r        G    xcexbd     (  xcexbb  )      (  cos  xcexb8  )         2    xcexbb    xcex93   (  xcexbb    1  xe2x88x95  2  )       xcfx80       (  xcexbd    xcexbb  )     xcexbb        cos   (   (  xcexbd    xcexbb  )   xcexb8  xe2x88x92  xcexbb  xcfx80  xe2x88x95  2  )       R    xcexbd     (  xcexbb  )      (  xcexb8  )   ,      and derive the precise expression of the xe2x80x9cresidualxe2x80x9d term       R    xcexbd     (  xcexbb  )      (  xcexb8  )      for all real     xcexbd  xe2x89xa5    xcexbd    0       (with some       xcexbd    0    u003e  0    ). With this at our disposal, we obtain the bounds of GGFFs uniform in     xcexbd  .     Under an appropriate weight function, the bounds are uniform for     xcexb8  xe2x88x88     0  ,  xcfx80        as well. Moreover, we can study the asymptotics of GGFFs with large fractional degree     xcexbd  .     The second is to present miscellaneous properties of GGFFs for better understanding of this family of useful special functions.

Asymptotic Analysis and Numerical Methods for Oscillatory Infinite Generalized Bessel Transforms with an Irregular Oscillator In this work, we perform a complete asymptotic analysis and the construction of affordable quadrature rules for a class of oscillatory infinite Bessel transform with a general oscillator. Especially in the presence of critical points, e.g., endpoints, zeros and stationary points, we first derive a series of useful asymptotic expansions in inverse powers of the frequency parameter omega . The resulting asymptotic expansions clarify the large omega  behavior of the transform and provide powerful tools for designing quadrature rules and conducting error analysis. As a consequence, efficient and affordable new modified Filontype methods for computing the transform numerically are proposed. Particularly, we carry out the rigorous error analysis and obtain asymptotic error estimates in inverse powers of omega . Numerical examples can confirm our analysis. The accuracy can be improved greatly by either adding more derivatives interpolation at endpoints or adding more interior nodes. Moreover, only using a small number of nodes and multiplicities, we can obtain the required accuracy level. For fixed number of nodes and multiplicities, the higher accuracy can be achieved with the larger values of omega .