Linear recurrences associated to rays in Pascal's triangle and combinatorial identities

Year, pages: 2014, 287 - 300

Our main purpose is to describe the recurrence relation associated to the sum of diagonal elements laying along a finite ray crossing Pascal's triangle. We precise the generating function of the sequence of described sums. We also answer a question of Horadam posed in his paper [Chebyshev and Pell connections, Fibonacci Quart. 43 (2005), 108–121]. Further, using Morgan-Voyce sequence, we establish the nice identity $F_n+1-iF_n=iⁿ∑_k=0ⁿ\binom{n+k}{2k}(-2-i)^k$ of Fibonacci numbers, where $i$ is the imaginary unit. Finally, connections to continued fractions, bivariate polynomials and finite differences are given.

Belbachir, H., Komatsu, T., Szalay, L. 2014. Linear recurrences associated to rays in Pascal's triangle and combinatorial identities. In Mathematica Slovaca, vol. 64, no.2, pp. 287-300. 0139-9918. DOI: https://doi.org/10.2478/s12175-014-0203-0

Belbachir, H., Komatsu, T., Szalay, L. (2014). Linear recurrences associated to rays in Pascal's triangle and combinatorial identities. Mathematica Slovaca, 64(2), 287-300. 0139-9918. DOI: https://doi.org/10.2478/s12175-014-0203-0