A Formalism for Merging Power Series in Solving Multinomial Equations: Master Series and Master Numbers

It is shown that all solutions of the canonical equation y^a = 1 + a*x*y^b with arbitrary complex exponents a, b (case m = 1) and of its limiting cases — the binomial root y = (1 + a*x)^(1/a) at b = 0, the exponential y = e^x at a = 0, b = 0, and the Lambert equation y = e^(x*y^b) at a = 0 — as well as the logarithms of these solutions (case m = 0), are expressed by a single power master series

M(m; a; b; x) = m + x + ((m − a + 2b)/2!) x^2 + ((m − a + 3b)(m − 2a + 3b)/3!) x^3 + …

For multivalued solutions beyond the radius of convergence, a geometric criterion for selecting the conjugate series (the same master series, but with a different set of parameters M(m; a′; b′; x′)) is proposed, ensuring branch continuity on the complex plane.

This representation of power series in terms of master numbers not only allows one to compute coefficients in degenerate cases where the gamma-function representation loses meaning, but also provides a transparent analytical algorithm for the merge operation on master series.

The method is illustrated on problems in gas dynamics and plasma physics, where it is demonstrated that physically realizable states require the application of conjugate series beyond the radius of convergence of the basic expansion.