Self-Similar Integral-Series Transformation Formulas and Their Applications in Special Function Expansions

In classical analysis, transforming continuous definite integrals into discrete infinite series serves as a fundamental technique for understanding special functions and analytic number theory. This paper proposes a unified theoretical framework based on self-similar integral-series transformations over the unit interval [0, 1]. Starting from basic geometric series expansions and polynomial kernel integrals, we generalize the integral structure by incorporating high-order attenuation factors. We rigorously establish and prove the exact transformation identity:

Integral_0^1 (x^(a-1) * (1 - x^c)^lambda / (1 + x^b)) dx = (1/c) * Sum_{n=0}^{infinity} (-1)^n * B((a + b*n)/c, lambda + 1)

where B(u, v) denotes the classical Beta function, and a, b, c, lambda are complex parameters satisfying appropriate convergence conditions. By examining limiting behavior and parameter reductions, we demonstrate that this transformation seamlessly reproduces classical logarithmic and rational series while systematically improving series convergence through higher-order polynomial denominators. Furthermore, we express these alternating Beta series in terms of generalized hypergeometric functions and investigate their analytic continuation properties. This framework provides an automated operator-based approach for converting continuous kernel integrals into discrete special function series without relying on contour integration.