Sums of squares of binomial coefficients, with applications to Picard-Fuchs equations

الأصل · EN

For a fixed integer N, and fixed numbers b₁,...,bₙ, we consider sequences, the nth term (aₙ) of which is the sum of the squares of the terms in the expansion of (b₁ +... + bₙ)ⁿ. In the case all bᵢ=1, we give a formula for a recurrence relation for the aₙ. Otherwise we give an algorithm for finding a recurrence relation. As an application, we give the Picard-Fuchs equations for certain families of elliptic curves.