confusion with the usual parenthesis notation for a derivative. Moreover, the following equation holds: which generalizes with Cauchy's formula to The derivative with respect to the second variable has the form, This is evident from the contour integral representation below. The generalized Laguerre polynomials obey the differential equation which may be compared with the equation obeyed by the kth derivative of the ordinary Laguerre polynomial, where for this equation only. In Sturm–Liouville form the differential equation is which shows that is an eigenvector for the eigenvalue . Orthogonality The generalized Laguerre polynomials are orthogonal over with respect to the measure with weighting function : which follows from If denotes the gamma distribution then the orthogonality relation can be written as The associated, symmetric kernel polynomial has the representations (Christoffel–Darboux formula) recursively Moreover, Turán's inequalities can be derived here, which is The following integral is needed in the quantum mechanical treatment of the hydrogen atom, Series expansions Let a function have the (formal) series expansion Then The series converges in the associated Hilbert space if and only if Further examples of expansions Monomials are represented as while binomials have the parametrization This leads directly to for the exponential function. The incomplete gamma function has the representation In quantum mechanics In quantum mechanics the Schrödinger equation for the hydrogen-like atom is exactly solvable by separation of variables in spherical coordinates. The radial part of the wave function is a (generalized) Laguerre polynomial. Vibronic transitions in the Franck-Condon approximation can also be described using Laguerre polynomials. Multiplication theorems Erdélyi gives the following two multiplication theorems Relation to Hermite polynomials The generalized Laguerre polynomials are related to the Hermite polynomials: where the are the Hermite polynomials based on the weighting function , the so-called "physicist's version." Because of this, the generalized Laguerre polynomials arise in the treatment of the quantum harmonic oscillator. Relation to hypergeometric functions The Laguerre polynomials may be defined in terms of hypergeometric functions, specifically the confluent hypergeometric functions, as where is the Pochhammer symbol (which in this case represents the rising factorial). Hardy–Hille formula The generalized Laguerre polynomials satisfy the Hardy–Hille formula where the series on the left converges for and . Using the identity (see generalized hypergeometric function), this can also be written as This formula is a generalization of the Mehler kernel for Hermite polynomials, which can be recovered from it by using the relations between Laguerre and