Rational coefficients of orthogonal decompositions of certain functions

Decompositions of many elementary and special functions into series by orthogonal polynomials have coefficients known explicitly. However, these coefficients are almost always irrational. Therefore, any numerical method gives these coefficients approximately when calculating in any arithmetic. This also applies to spectral methods that provide efficient approximations of holonomic functions. However, in some exceptional cases, the expansion coefficients obtained by the spectral method turn out to be rational and are calculated exactly in rational arithmetic. We consider such decompositions with respect to some classical orthogonal polynomials. It is shown that in this way it is possible to obtain an infinite set of linear forms for some irrationalities, in particular, for Euler’s constant.