Legendre Transform
المؤلف:
Jin, Y. and Dickinson, H.
المصدر:
"Apéry Sequences and Legendre Transforms." J. Austral. Math. Soc. Ser. A 68,
الجزء والصفحة:
...
29-10-2020
876
Legendre Transform
The Legendre transform of a sequence ![<span style=]()
{c_k}" src="https://mathworld.wolfram.com/images/equations/LegendreTransform/Inline1.gif" style="height:15px; width:22px" /> is the sequence ![<span style=]()
{a_k}" src="https://mathworld.wolfram.com/images/equations/LegendreTransform/Inline2.gif" style="height:15px; width:23px" /> with terms given by
where 
is a binomial coefficient (Jin and Dickinson 2000, Zudilin 2004). The inverse Legendre transform is then given by
 |
(3)
|
where
(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.
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