Can someone help me with this question? The topic is PDE AND WAVE

Orthogonal polynomials Consider the g-erating function 1 m n 1 2tz+t2 : EMS}: ' (1') uestion 3 a : U ' nation 7 , nd a. mild-order dierential nation of the form, Q { ) 5mg Bq eq 11(3):,4311 + 9(1)H:-.(Il + 1\" yak) = 0- (3) Where p[:r} and (pit) are polynomials in :r (at most 2nd degree]1 and A" is an eigenvalue that depends on 1:. What are plzz), q{:r:), and A\"? Question 30)): Prove that 1 Lymummwtzldx = Purim\". it man. (9) for some weight function 10(1), and normalisation 1",... What are w(z} and 1""? Question 3(a): Use the generating function to nd a. three-term recurrence relation of the form, anyn+1(r} + bnyntr) + Gums1(1) = I yntr) (10) Find on, b," and on. Question 3(d): Prove that 911(1) = ZH(I)P{I} {11) [=0 Where 3(3) are the Legendre polynomials. Hint: this question is easy if you use a specic property of generating mctions