Question

Laguerre's equation ty" + (1 – 2)y' + py=0) where p is a constant, is a confluent hypergeometric equation. (a) Show that the equation has x = 0) as a regular singular point with indicial root 0; (b) Show that one solution of the equation is 1+ Ž (–1)n-P(p – 1) ...(p+1-n) n=1 (n!) (©) Show that the only solutions bounded near the origin are constant multiples of the solution in (b).

Answer 1

Consider laguerre's ea Xy"_*(1-x1y't py-o Where pis 9 -1)) constant is a Confluent hypergeometni ez? Dividing ci) throughout by a y" + (1-x) on-analytic at x=0 , and and xalxi= px Which are. This makes x=0 a regular singularity with Po = lim (1-x)=1 (1-x) y' + @Jy =0 Which are x non XP(X) = (1-x) xto and tim px = 0 na Therefore eq".(1) has x=0 as a regular singular pt. with indicial root o. We seek a form of a Frobenius series: ус) o anants anzo if nco (b) Ne-0 nts-1 ... y' = 5 (ntsian y"=5 (nts)(ntsu) anx no- no ents-2 ..2 (nts)(n+51) 9n 2 +(1-x) Ents) an X + p { ana n2-00 nos -0 ク:10 nu-1 . Ž (onts) ints-1) anants-1 + (nts) and no-00 I (nts) anx no- NTS + P E anx o 7:00 ns-

Relabeling indices È Einarest ents) anti tf r4178-7-7-pJau] And the recurrence relation is : (19SKAA cntits) (nts) and + (1-P) 9220 Setting na-1 yields (92=0,980) the indicial ean s (5-1) = 0 either szo For s=o anti -(1-P) an 2-1 an incnti) nanti) to nal ar for n=1 a2 = 1 2 ar) p=3 ay = au na 2 935 Pay az =P- 2 (2+) 2-3 P-) az cp-10² 3 (3+1) 3-4 22.3 y= aorta,x + a2x² + 93x²+ aux"+.. = got axt 3 2.3 (P-1) प aj + 2.3-4 2 41 x+ 2 qa 2 art tspunten + ..(prin) n (n!) one sol is It Š FI)" (P-1) x M21