<abstract xmlns="http://www.w3.org/1999/xhtml">In 1997, Richard André-Jeannin obtained a symmetric identity involving the reciprocal of the Horadam numbers <italic>Wn</italic>, defined by a three-term recurrence <italic>Wn</italic>+2 = <italic>P Wn</italic>+1<italic>− QWn</italic> with constant coefficients. In this paper, we extend this identity to sequences <italic>{an}n</italic>∈ℕ satisfying a three-term recurrence <italic>an</italic>+2 = <italic>pn</italic>+1<italic>an</italic>+1 + <italic>qn</italic>+1<italic>an</italic> with arbitrary coefficients. Then, we specialize such an identity to several <italic>q</italic>-polynomials of combinatorial interest, such as the <italic>q</italic>-Fibonacci, <italic>q</italic>-Lucas, <italic>q</italic>-Pell, <italic>q</italic>-Jacobsthal, <italic>q</italic>-Chebyshev and <italic>q</italic>-Morgan-Voyce polynomials.</abstract>

In 1997, Richard André-Jeannin obtained a symmetric identity involving the reciprocal of the Horadam numbers Wn, defined by a three-term recurrence Wn+2 = P Wn+1− QWn with constant coefficients. In this paper, we extend this identity to sequences {an}n∈ℕ satisfying a three-term recurrence an+2 = pn+1an+1 + qn+1an with arbitrary coefficients. Then, we specialize such an identity to several q-polynomials of combinatorial interest, such as the q-Fibonacci, q-Lucas, q-Pell, q-Jacobsthal, q-Chebyshev and q-Morgan-Voyce polynomials.

A generalization of André-Jeannin’s symmetric identity

Pure Mathematics and Applications

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{"article-title":"A generalization of André-Jeannin\u2019s symmetric identity"}

In 1997, Richard André-Jeannin obtained a symmetric identity involving the reciprocal of the Horadam numbers Wn, defined by a three-term recurrence Wn+2 = P...

A generalization of André-Jeannin’s symmetric identity

Published Online: Aug 06, 2018

Page range: 98 - 118

Received: Oct 30, 2017

DOI: https://doi.org/10.1515/puma-2015-0028

Keywords
combinatorial sums, sums of reciprocals, three-term recurrences, -Fibonacci polynomials, -Fibonacci numbers, -Lucas polynomials, -Lucas numbers, -Pell polynomials, -Pell numbers, -Jacobsthal polynomials, -Jacobsthal numbers, -Chebyshev polynomials, -Morgan-Voyce polynomials

© 2018 Emanuele Munarini, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

A generalization of André-Jeannin’s symmetric identity

Published Online: Aug 06, 2018

Page range: 98 - 118

Received: Oct 30, 2017

DOI: https://doi.org/10.1515/puma-2015-0028

Keywordscombinatorial sums, sums of reciprocals, three-term recurrences, -Fibonacci polynomials, -Fibonacci numbers, -Lucas polynomials, -Lucas numbers, -Pell polynomials, -Pell numbers, -Jacobsthal polynomials, -Jacobsthal numbers, -Chebyshev polynomials, -Morgan-Voyce polynomials

© 2018 Emanuele Munarini, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
combinatorial sums, sums of reciprocals, three-term recurrences, -Fibonacci polynomials, -Fibonacci numbers, -Lucas polynomials, -Lucas numbers, -Pell polynomials, -Pell numbers, -Jacobsthal polynomials, -Jacobsthal numbers, -Chebyshev polynomials, -Morgan-Voyce polynomials