An application of the zeros of Laguerre polynomials

A theorem is formulated and substantiated numerically, stating that if λ_{m, n}^(v) is the n th zero of the generalized Laguerre polynomial L_m^(v)(X) (x and λ_{m, n}^(v) - real, positive, v =0,1,2,..., m =0,1,2,..., n =1,2,3...) and L(c, n) (c =1,2,3,...) is a positive real number, defined by means of the relation: L(c, n) =_{k_→-∞}^lim K_(c, n, k_) = _{k_→-∞}^lim M_(c, n, k_) in which K_(c, n, k_) = |k_|ζ^(c)_{k_, n}, M_(c, n, k_) = |a_|ζ^(c)_{k_, n}, and ζ^(c)_{k_, n} is the n th positive purely imaginary zero in x of the Kummer confluent hypergeometric function Φ(a_, c; x) with a_= c/2-jk_ - complex, x =jz, z - real, positive and k_ - real, negative, (c, n - fixed), in case v =c -1 and m is large, it holds: λ_{m, n}^(v) ≈L(c, n). The result obtained is used to develop a simple approximate method for computation of the differential phase shift, provided by the azimuthally magnetized circular ferrite waveguide for the normal TE₀₁ mode.