Rational coefficients of orthogonal expansions of some functions

Expansions of many elementary and special functions in series of orthogonal polynomials have coefficients known explicitly. However, almost always these coefficients are irrational. Thus any numerical method would give these coefficients approximately with computations in any arithmetic. This is also true for the spectral methods, which give effective approximations for holonomic functions. However, in some exceptional cases the coefficients of expansions obtained with spectral methods turn out to be rational, and are computed exactly in rational arithmetic. We consider these expansions in series of some classical orthogonal polynomials. We demonstrate that, in this way, an infinite number of linear forms can be obtained for some irrationalities, in particular, for the Euler constant.