A Hypergeometric Basis for the Alpert Multiresolution Analysis

We construct an explicit orthonormal basis of piecewise _{i+1}F_{i}$ hypergeometric polynomials for the Alpert multiresolution analysis. The Fourier transform of each basis function is written in terms of _2F_3$ hypergeometric functions. Moreover, the entries in the matrix equation connecting the wavelets with the scaling functions are shown to be balanced _4 F_3$ hypergeometric functions evaluated at $1$, which allows us to compute them recursively via three-term recurrence relations. The above results lead to a variety of new interesting identities and orthogonality relations reminiscent of classical identities of higher-order hypergeometric functions and orthogonality relations of Wigner 6j-symbols.