Bessel Polynomial

Author

Eric W. Weisstein
December 19, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/BesselPolynomial.nb.

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

©2009 Wolfram Research, Inc. except for portions noted otherwise

Definition (Subscript[y, n])

Sum

Table[Sum[(((n+k)!)/((n-k)!k!))((((x/2)))^k),{k,0,n}],{n,0,5}]

(CoefficientList[#,x]&/@Table[Sum[(((n+k)!)/((n-k)!k!))((((x/2)))^k),{k,0,n}],{n,0,5}])

Sum[(((n+k)!)/((n-k)!k!))((((x/2)))^k),{k,0,n}]//Simplify

Table

Table[(E^(1/x)) (Sqrt[(2/π)]) (Sqrt[(1/x)]) BesselK[-(1/2)-n,(1/x)],{n,0,5}]//TableForm//TraditionalForm

Flatten[Reverse/@(CoefficientList[#,x]&/@Table[(E^(1/x)) (Sqrt[(2/π)]) (Sqrt[(1/x)]) BesselK[-(1/2)-n,(1/x)],{n,0,5}])]

Differential Equation

DSolve[x^2y''[x]+(2x+2)y'[x]-n(n+1)y[x]==0,y[x],x]//FullSimplify

FullSimplify[x^2y''[x]+(2x+2)y'[x]-n(n+1)y[x]/.y->Function[x,(E^(1/x)) (Sqrt[(2/π)]) (Sqrt[(1/x)]) BesselK[-(1/2)-n,(1/x)]]]

Plot

Plot[Evaluate[Table[(E^(1/x)) (Sqrt[(2/π)]) (Sqrt[(1/x)]) BesselK[-(1/2)-n,(1/x)],{n,0,5}]],{x,-1.2,.5},PlotStyle->List/@{Black,Red,Orange,Green,Blue,Purple},
PlotRange->{-1,2},AxesLabel->TraditionalForm/@{x,(Subscript[y, n])[x]}]

Definition (Subscript[p, n])

Generating Function

Factor/@Series[Exp[x(1-(Sqrt[1-2t]))],{t,0,4}]

Table[Sum[(((2n-k-1)!)/((2^(n-k))(k-1)!(n-k)!))(x^k),{k,n}],{n,0,4}]

Reverse/@(CoefficientList[#,x]&/@Table[Sum[(((2n-k-1)!)/((2^(n-k))(k-1)!(n-k)!))(x^k),{k,n}],{n,0,5}])

Sum

V5

Sum[(((2n-k-1)!)/((2^(n-k))(k-1)!(n-k)!))(x^k),{k,n}]

...wrong...

Table[%,{n,0,4}]//FullSimplify

V8

$Version

Sum[((2*n-k-1)!/(2^(n-k)*(k-1)!*(n-k)!))*x^k,{k,n}]

FullSimplify[%,n∈Integers&&n>0]

Table[(((2^(-1+n)) x Gamma[-(1/2)+n] Hypergeometric1F1[1-n,2-2 n,2 x])/(Sqrt[π])),{n,5}]//Expand

FunctionExpand[%3,n∈Integers&&n>0]

{(2n-1),2^n}/.n->n-1//Simplify

FullSimplify[(((2^(-1+n)) x Gamma[-(1/2)+n] Hypergeometric1F1[1-n,2-2 n,2 x])/(Sqrt[π]))/.Gamma[n-1/2]->(((2n-3)!!)/(2^(n-1)))(Sqrt[π])]

Table[x (-3+2 n)!! Hypergeometric1F1[1-n,2-2 n,2 x],{n,5}]//Expand

Differential Equation

DSolve[y''[x]-2y'[x]+2n y[x]==0,y[x],x]

Plot

Plot[Evaluate[Table[Sum[(((2n-k-1)!)/((2^(n-k))(k-1)!(n-k)!))(x^k),{k,n}],{n,5}]],{x,-2,2},PlotStyle->List/@{Red,Orange,Green,Blue,Purple},
PlotRange->{-6,6},AxesLabel->TraditionalForm/@{x,(Subscript[p, n])[x]}]