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Legendre Transform

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The Legendre transform of a sequence {c_k} is the sequence {a_k} with terms given by

a_n=sum_(k=0)^(n)(n; k)(n+k; k)c_k
(1)
=sum_(k=0)^(n)(2k; k)(n+k; n-k)c_k,
(2)

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

(2n; n)c_n=sum_(k=0)^n(-1)^(n-k)d_(n,k)a_k,
(3)

where

d_(n,k)=(2n; n-k)-(2n; n-k-1)
(4)
=(2k+1)/(n+k+1)(2n; n-k)
(5)

(Zudilin 2004).

Strehl (1994) and Schmidt (1995) showed that

sum_(k=0)^n(n; k)^2(n+k; k)^2=sum_(k=0)^n(n; k)(n+k; k)sum_(j=0)^k(k; j)^3.
(6)

See also

Binomial Sums, Legendre Transformation, Schmidt's Problem, Strehl Identities

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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://doi.org/10.37236/1775.

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Legendre Transform

Cite this as:

Weisstein, Eric W. "Legendre Transform." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LegendreTransform.html

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