I have been playing around with the theory of orthogonal polynomials, and it occurred to me that we might be able to build a family of orthogonal polynomials from even powers of a variable $x$. For concreteness let's consider moments of the following measure:
$$m_{n} = \int_{0}^{\infty}\frac{x^{2n}}{\cosh\left(\pi x/2\right)}dx $$
With this definition, it is natural to try and construct an orthonormal basis of polynomials on the positive real axis via determinants, i.e.,
$$P_{n}(x) = c_{n}\text{det}\begin{pmatrix} m_{0} && m_{1} && m_{2} && \cdots && m_{n}\\ m_{1} && m_{2} && m_{3} && \cdots && m_{n+1}\\ \vdots && \vdots && \vdots && \ddots && \vdots \\ m_{n-1} && m_{n} && m_{n+1} && \cdots && m_{2n-1}\\ 1 && x^{2} && x^{4} && \cdots && x^{2n} \end{pmatrix}$$
where $c_{n}$ is chosen such that
$$\int_{0}^{\infty}\frac{P_{n}(x)^{2}}{\cosh(\pi x/2)} = 1$$
I've computed the first 20 polynomials numerically, and confirmed that these are indeed orthonormal. What I would like to do is set up a recurrence relation, similar in spirit to Favard's theorem, which allows for the derivation of subsequent polynomials from the previous ones and hopefully proves that the full sequence of polynomials is orthonormal. I would appreciate direction towards such a theorem, or any resources which explain how to derive such recurrence relations in general (this subject is very new to me).
Edit: As mentioned in a comment by Carl, the moments defined above are precisely the (even) Euler numbers, or secant numbers. That is why I was motivated to look at this sequence, and it is already known that there is a set of orthonormal polynomials for the full sequence of Euler numbers. These polynomials aren't strictly even or odd, so there isn't a clear relation to me between those polynomials and the ones I am looking for here.
Edit 2: I have made more progress by experimenting with low-order polynomials. It seems that (at least for the first 20 polynomials) that there is a recursion relation of the form
$$P_{n+1}(x) = (a_{n}+b_{n}x^{2})P_{n}(x) + c_{n}P_{n-1}(x)$$
which is analogous to the standard form in Favard's theorem and related literature. I'm hoping to prove that this holds to all orders and that the full set of polynomials is therefore orthogonal. Moreover I'd like to find a generating function for them.