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Abstract

This paper explicitly constructs the integral kernels for a family of weight-shifting operators mapping from the space of weight-k smooth automorphic functions to the space of weight-t automorphic functions. These operators are defined as integral transforms whose kernel, K, is the product of a covariant factor P and a G-invariant function F. The key assumption in the kernel's construction is the spectral condition that K is an eigenfunction of the weight-t hyperbolic Laplacian, Delta_t. This partial differential equation reduces to an ordinary differential equation for the invariant function F, which is shown to be equivalent to the Hypergeometric Differential Equation (HDE). The main result of this paper is the explicit determination of the HDE parameters (a,b,c) as functions of the automorphic data (k,t,q) and the spectral parameter lambda_K. As a foundational step toward the rigorous realization of the operator, we conclude with an analysis of the absolute convergence of the corresponding integral transform. This analysis establishes a region where the operator is well-defined and frames the problem of its global definition via analytic continuation, a task deferred to a subsequent investigation.