## Christian Krattenthaler

# Hankel determinants of linear combinations
of moments of orthogonal polynomials, II

### (28 pages)

**Abstract.**
We present a formula that expresses the Hankel determinants of
a linear combination of length *d*+1 of moments of orthogonal polynomials
in terms of a *d* x *d* determinant of the orthogonal polynomials.
This formula exists somehow hidden in the folklore of the theory of
orthogonal polynomials but deserves to be better known, and be
presented correctly and with full proof. We present three
fundamentally different
proofs, one that uses classical formulae from the theory of orthogonal
polynomials, one that uses a vanishing argument and is due to
Elouafi [*J. Math. Anal. Appl.* **431** (2015), 1253-1274]
(but given in an incomplete form), and one that uses (Dodgson) condensation.
We give two applications of the formula. In the first application,
we explain how to
compute such Hankel determinants in a singular case. The second
application concerns the linear recurrence of such Hankel
determinants for a certain class of moments that covers numerous
classical combinatorial sequences, including Catalan numbers,
Motzkin numbers,
central binomial coefficients, central trinomial coefficients,
central Delannoy numbers, Schr\"oder numbers, Riordan numbers,
and Fine numbers.

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