Legendre Transform

The Legendre transform of a sequence  is the sequence  with terms given by

			(1)
			(2)

where  is a binomial coefficient (Jin and Dickinson 2000, Zudilin 2004). The inverse Legendre transform is then given by

(3)

where

			(4)
			(5)

(Zudilin 2004).

Strehl (1994) and Schmidt (1995) showed that

(6)

REFERENCES:

Jin, Y. and Dickinson, H. "Apéry Sequences and Legendre Transforms." J. Austral. Math. Soc. Ser. A 68, 349-356, 2000.

Schmidt, A. L. "Legendre Transforms and Apéry's Sequences." J. Austral. Math. Soc. Ser. A 58, 358-375, 1995.

Strehl, V. "Binomial Identities--Combinatorial and Algorithmic Aspects. Trends in Discrete Mathematics." Disc. Math. 136, 309-346, 1994.

Zudilin, W. "On a Combinatorial Problem of Asmus Schmidt." Elec. J. Combin. 11, R22, 1-8, 2004. https://www.combinatorics.org/Volume_11/Abstracts/v11i1r22.html.