A system of 4 simultaneous recursions: Generalization of Ledin-Shannon-Ollerton

Abstract: This paper further generalizes a recent result of Shannon and Ollerton who resurrected an old identity due to Ledin. This paper generalizes the Ledin-Shannon-Ollerton result to all metallic sequences. The results give closed formulas for the sum of products of powers of the first $n$ integers with the first $n$ members of the metallic sequence. Three key innovations of this paper are (i) reducing the proof of the generalization to the solution of a system of 4 simultaneous recursions; (ii) skillful use of the shift operation to prove equality of polynomials; and (iii) new OEIS sequences arising from the coefficients of the four polynomial families satisfying the four simultaneous recursions.

number theory

Audience: researchers in the discipline

( paper )

Combinatorial and additive number theory (CANT 2022)