Complete solutions of inverse quantum orthogonal equivalence classes

Abstract

In the year 1939, the mathematician G.H. Hardy proved that the only functions f which satisfy the classical orthogonality relation ∫01f(λmt)f(λnt)dt=0,m≠n,are the Bessel functions Jν(t) under certain constraints, where ν>−1 is the order of the Bessel function, and λm, λn are the nontrivial zeros of the Bessel function. By example, herein we present a complete equivalence class of inverse q-periodic functions, satisfying the inverse q-orthogonality relation ∫01f(λmt)f(λnt)d1qt=0,m≠n.The inverse q-integral is a Riemann–Stieltjes integral with respect to a step function having infinitely many points of increase at the points q−ℓ, with the step size at the point q−ℓ being q, ∀ℓ∈N0, where N0≔N∪{0}, and |q|≠1. An equivalence class of entire functions f∈Lq−12(0,1) that are inverse q-orthogonal with respect to their own nontrivial zeros is proven by invoking the Fundamental Theorem of Algebra, as the inverse q-periodic functions constitute an equivalence class of nonzero inverse q-periodic constants on the complex plane.