#####
Séminaire Lotharingien de Combinatoire, B29b (1992), 22
pp.

[Formerly: Publ. I.R.M.A. Strasbourg, 1993, 1993/033, p.
81-100.]

# Volker Strehl

# Recurrences and Legendre Transform

**Abstract.**
A binomial identity which relates to the famous Apéry
numbers and the sums of cubes of binomial coefficients (for
which Franel has established a recurrence relation almost one
hundred years ago) can be seen as a particular instance of a
Legendre transform between sequences. A proof of this
identity can be based on the more general fact that the
Apéry and Franel recurrence relations themselves are
conjugate via Legendre transform. This motivates a closer look
at conjugacy of sequences satisfying linear recurrence
relations with polynomial coefficients. The role of
computer-aided proof and verification in the study of
binomial identities and recurrence relations is illustrated,
and potential applications of conjugacy in diophantine
approximation are mentioned.

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