Differential Equations and Linear Algebra (4th Edition)

by Goode, Stephen W.; Annin, Scott A.

Published by Pearson

ISBN 10: 0-32196-467-5

ISBN 13: 978-0-32196-467-0

Chapter 11 - Series Solutions to Linear Differential Equations - 11.6 Bessel's Equation of Order p - Problems - Page 783: 4

Answer

See below

Work Step by Step

Let $\Gamma (p)$ denote the gamma function. Obtain: $p \Gamma (p)=\Gamma (p+1)$ then $\Gamma (p+k+1)=(p+k)\Gamma (p+k)\\ =(p+k)\Gamma (p+k-1+1)\\ =(p+k)\Gamma (p+k-1+1)\\ =(p+k)(p+k-1)\Gamma (p+k-1)\\ =(p+k)(p+k-1)\Gamma (p+k-2+1)\\ =(p+k)(p+k-1)(p+k-2)\Gamma (p+k-2)\\ =(p+k)(p+k-1)(p+k-2)...(p+3)\Gamma (p+3)\\ =(p+k)(p+k-1)(p+k-2)... (p+3)(p+2)\Gamma (p+2)\\ =(p+k)(p+k-1)(p+k-2)...(p+3)(p+2)\Gamma (p+1+1)\\ =(p+k)(p+k-1)(p+k-2)...(p+3)(p+2)(p+1)\Gamma (p+1)$

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