Indicial Equation

Author

Eric W. Weisstein
October 10, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DifferentialEquations/IndicialEquation.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/IndicialEquation.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Bessel

v4.2

<<DiscreteMath`RSolve`

RSolve[{a[n]n(2m+n)+a[n-2]==0,a[0]==a0,a[1]==0},a[n],n]

v5.2

RSolve[{a[n]n(2m+n)+a[n-2]==0,a[0]==a0,a[1]==0},a[n],n]

Chebyshev

v4.2

RSolve[{(n+2)(n+1)a[n+2]+((α^2)-(n^2))a[n]==0,a[0]==a0,a[1]==a1},a[n],n]

v5.2

RSolve[{(n+2)(n+1)a[n+2]+((α^2)-(n^2))a[n]==0,a[0]==a0,a[1]==a1},a[n],n]

Hermite

v4.2

RSolve[{(n+2)(n+1)a[n+2]+(λ-2n)a[n]==0,a[1]==a1,a[0]==a0},a[n],n]

v5.2

RSolve[{(n+2)(n+1)a[n+2]+(λ-2n)a[n]==0,a[1]==a1,a[0]==a0},a[n],n]

Jacobi

v4.2

RSolve[{n(n+α+β+1)-ν(ν+α+β+1)a[ν]-2(ν+1)(ν+α+1)a[ν+1]==0,a[0]==a0},a[ν],ν]

v5.2

RSolve[{n(n+α+β+1)-ν(ν+α+β+1)a[ν]-2(ν+1)(ν+α+1)a[ν+1]==0,a[0]==a0},a[ν],ν]

Laguerre

v4.2

RSolve[{(n+1)(n+ν+1)a[n+1]+(λ-n)a[n]==0,a[0]==a0},a[n],n]//FullSimplify

v5.2

RSolve[{(n+1)(n+ν+1)a[n+1]+(λ-n)a[n]==0,a[0]==a0},a[n],n]//FullSimplify

Resummation

Sum[((a0 Gamma[n-λ] Gamma[1+ν])/(Gamma[1+n] Gamma[-λ] Gamma[1+n+ν]))(x^n),{n,0,∞}]

Series

Series[Hypergeometric1F1[-λ,1+ν,x],{x,0,4}]

Special Case

With[{ν=0},Hypergeometric1F1[-λ,1+ν,x]]//FunctionExpand

FunctionExpand[Hypergeometric1F1[-λ,1+ν,x],ν∈Integers&&ν>0]

Table[Hypergeometric1F1[-λ,1+ν,x],{λ,0,3}]

Factor[Factor/@%]

Factor/@Factor[Table[((λ!LaguerreL[λ,ν,x])/(Pochhammer[ν+1,λ])),{λ,0,3}]]

FullSimplify[((λ!LaguerreL[λ,ν,x])/(Pochhammer[ν+1,λ]))]

Legendre

v4.2

RSolve[{(n+1)(n+2)a[n+2]+(-n(n+1)+l(l+1))a[n]==0,a[0]==a0,a[1]==a1},a[n],n]

v5.2

RSolve[{(n+1)(n+2)a[n+2]+(-n(n+1)+l(l+1))a[n]==0,a[0]==a0,a[1]==a1},a[n],n]