A. Show that w=e^(x^(2/4))y is a solution of Weber\'s equation, w\"+(n+(1/2)-(1/
ID: 3087117 • Letter: A
Question
A. Show that w=e^(x^(2/4))y is a solution of Weber's equation, w"+(n+(1/2)-(1/4)x^2)w=0, where (n=0,1,2,...). B. Show that the following product converges, and find its value: pi, evaluated from n=1 to infinity, [1+6/(n+1)(2n+9)]. C. Show that: pi, evaluated from n=2 to infinity, (1-1/n^2)=(1/2).Explanation / Answer
For arbitrary real a the polynomial solution of the differential equation [1] is called a generalized Laguerre polynomials, or associated Laguerre polynomials. The Rodrigues' formula for them are The simple Laguerre polynomials are recovered from the generalized polynomials by setting a = 0: [edit]Explicit examples and properties of generalized Laguerre polynomials Laguerre functions are defined by confluent hypergeometric functions and Kummer's transformation as[2] When n is an integer the function reduces to a polynomial of degree n. It has the alternative expression[3] in terms of Kummer's function of the second kind. The generalized Laguerre polynomial of degree n is[4] (derived equivalently by applying Leibniz's theorem for differentiation of a product to Rodrigues' formula.) The first few generalized Laguerre polynomials are: The coefficient of the leading term is (-1)n/n!; The constant term, which is the value at 0, is Ln(a) has n real, strictly positive roots (notice that is a Sturm chain), which are all in the interval [citation needed] The polynomials' asymptotic behaviour for large n, but fixed a and x > 0, is given by[citation needed] and summarizing by where is the Bessel function. Moreover[citation needed] whenever n tends to infinity. [edit]Recurrence relations This section does not cite any references or sources. (September 2011) The addition formula for Laguerre polynomials[5]: . Laguerre's polynomials satisfy the recurrence relations in particular and or moreover They can be used to derive the four 3-point-rules combined they give this additional, useful recurrence relations A somewhat curious identity, valid for integer i and n, is it may be used to derive the partial fraction decomposition