Morgan-Voyce Polynomials

Author

Eric W. Weisstein
December 13, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/Morgan-VoycePolynomials.nb.

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Code

MorganVoyceB[n_,x_]:=Module[{k},Sum[Binomial[n+k+1,n-k](x^k),{k,0,n}]]
MorganVoyceb[n_,x_]:=Module[{k},Sum[Binomial[n+k,n-k](x^k),{k,0,n}]]

Plots

Show[GraphicsArray[{plot1,plot2}],GraphicsSpacing->0,ImageSize->500]

Sums

(Subscript[B, n])(x)==(∑_(k=0)^n)(x^k) (Binomial[k+n+1, n-k])

V6

Sum[Binomial[n+k+1,n-k](x^k),{k,0,n}]

FullSimplify[%,n≥0&&n∈Integers]

V7.0.0

FullSimplify[Sum[Binomial[n+k+1,n-k](x^k),{k,0,n}],x>0&&n>0&&n∈Integers]

Table[Subtract@@{((2 Sinh[2 (1+n) ArcCsch[(2/(Sqrt[x]))]])/((Sqrt[x]) (Sqrt[4+x]))),(1/(Sqrt[x (4+x)]))(2^(-1-n)) (-(2+x) (((2+x-(Sqrt[x (4+x)])))^n)+(Sqrt[x (4+x)]) (((2+x-(Sqrt[x (4+x)])))^n)+(2+x) (((2+x+(Sqrt[x (4+x)])))^n)+(Sqrt[x (4+x)]) (((2+x+(Sqrt[x (4+x)])))^n))},{n,0,10}]//FullSimplify//Union

V8

Sum[Binomial[n+k+1,n-k](x^k),{k,0,n}]

Fibonacci V7.0.1

FullSimplify[((2 Sinh[2 (1+n) ArcCsch[(2/(Sqrt[x]))]])/((Sqrt[x]) (Sqrt[4+x])))-FunctionExpand@Fibonacci[2n+2,(Sqrt[x])]/(Sqrt[x]),n∈Integers]

Fibonacci V8

FullSimplify[((2 Sinh[2 (1+n) ArcCsch[(2/(Sqrt[x]))]])/((Sqrt[x]) (Sqrt[4+x])))-FunctionExpand@Fibonacci[2n+2,(Sqrt[x])]/(Sqrt[x]),n∈Integers]

{$Version,$ReleaseNumber}

Table[((2 Sinh[2 (1+n) ArcCsch[(2/(Sqrt[x]))]])/((Sqrt[x]) (Sqrt[4+x])))-Fibonacci[2n+2,(Sqrt[x])]/(Sqrt[x])/.{n->RandomInteger[{-10,10}],x->RandomComplexDisk[]},{10}]//Chop//Union

(Subscript[b, n])(x)==(∑_(k=0)^n)(x^k) (Binomial[k+n, n-k])

V6

Sum[Binomial[n+k,n-k](x^k),{k,0,n}]

FullSimplify[%,n≥0&&n∈Integers]

V7

FullSimplify[Sum[Binomial[n+k,n-k](x^k),{k,0,n}],x∈Integers&&n>0&&n∈Integers]

Table[Subtract@@{((2 Cosh[(1+2 n) ArcCsch[(2/(Sqrt[x]))]])/(Sqrt[4+x])),(1/(Sqrt[4+x]))(2^(-1-n)) ((Sqrt[x]) (-(((2+x-(Sqrt[x]) (Sqrt[4+x])))^n)+(((2+x+(Sqrt[x]) (Sqrt[4+x])))^n))+(Sqrt[4+x]) ((((2+x-(Sqrt[x]) (Sqrt[4+x])))^n)+(((2+x+(Sqrt[x]) (Sqrt[4+x])))^n)))},{n,0,10}]//FullSimplify//Union

V8

Sum[Binomial[n+k,n-k](x^k),{k,0,n}]

Fibonacci V7

FullSimplify[((2 Cosh[(1+2 n) ArcCsch[(2/(Sqrt[x]))]])/(Sqrt[4+x]))-FunctionExpand@Fibonacci[2n+1,(Sqrt[x])],n∈Integers]

Table[((2 Cosh[(1+2 n) ArcCsch[(2/(Sqrt[x]))]])/(Sqrt[4+x]))-Fibonacci[2n+1,(Sqrt[x])]/.{n->RandomInteger[{-10,10}],x->RandomComplexDisk[]},{10}]//Chop//Union

Fibonacci V8

FullSimplify[((2 Cosh[(1+2 n) ArcCsch[(2/(Sqrt[x]))]])/(Sqrt[4+x]))-FunctionExpand@Fibonacci[2n+1,(Sqrt[x])],n∈Integers]

(Subscript[B,  n])(x)

Plots

plot1=Plot[Evaluate[Table[MorganVoyceB[n,x],{n,0,5}]],{x,-2,1},
PlotRange->{-2,10},
AxesLabel->TraditionalForm/@{x,HoldForm[(Subscript[B, n])[x]]},
PlotStyle->{Black,Red,Orange,Yellow,Green,Blue}]

Analytic

B=FullSimplify[MorganVoyceB[n,x],x>0&&n>0&&n∈Integers]

In Terms of θ and ϕ

FullSimplify[B/.x->2Cos[θ]-2,0<t<Pi/2]

FullSimplify[TrigExpand/@(Divide@@@Table[{B/.x->(2Cos[t]-2),Sin[(n+1)t]/Sin[t]},
{n,10}]),0<t<Pi/2]

FullSimplify[B/.x->2(Cosh[ϕ]-1),ϕ>0]//Simplify

In Terms of Q

Q={{x+2,-1},{1,0}};

MatrixForm/@Simplify[NestList[Dot[Q,#]&,Q,5]]

In Terms of Fibonacci Polynomials

FullSimplify[FunctionExpand[Fibonacci[2n+2,x]/x-MorganVoyceB[n,x^2]],n∈Integers]

DiffyQ

x(x+4)y''[x]+3(x+2)y'[x]-n(n+2)y[x]/.y->Function[x,Evaluate[MorganVoyceB[n,x]]]//FullSimplify

DSolve[x(x+4)y''[x]+3(x+2)y'[x]-n(n+2)y[x]==0,y[x],x]

FullSimplify[%,n∈Integers]

(Subscript[b,  n])(x)

Plots

plot2=Plot[Evaluate[Table[MorganVoyceb[n,x],{n,0,5}]],{x,-2,1},
PlotRange->{-2,10},
AxesLabel->TraditionalForm/@{x,HoldForm[(Subscript[b, n])[x]]},
PlotStyle->{Black,Red,Orange,Yellow,Green,Blue}]

Analytic

b=FullSimplify[MorganVoyceb[n,x],x>0&&n>0&&n∈Integers]

In Terms of θ and ϕ

b=FullSimplify[MorganVoyceb[n,x],x>0&&n>0&&n∈Integers]

FullSimplify[b/.x->2Cos[θ]-2,0<t<Pi/2]

FullSimplify[TrigExpand/@(Divide@@@Table[{b/.x->(2Cos[t]-2),Cos[(2n+1)t/2]/Cos[t/2]},
{n,10}]),0<t<Pi/2]

FullSimplify[b/.x->2(Cosh[ϕ]-1),ϕ>0]//Simplify

In Terms of Fibonacci Polynomials

FullSimplify[FunctionExpand[Fibonacci[2n+1,x]-MorganVoyceb[n,x^2]],n∈Integers]

Identities

{MorganVoyceb[0,x],MorganVoyceB[0,x]}

FullSimplify[Subtract@@{MorganVoyceb[n,x],x MorganVoyceB[n-1,x]+MorganVoyceb[n-1,x]}]

FullSimplify[Subtract@@{MorganVoyceB[n,x],(x+1)MorganVoyceB[n-1,x]+MorganVoyceb[n-1,x]}]

FullSimplify[Subtract@@{MorganVoyceb[n,x],(x+2)MorganVoyceb[n-1,x]-MorganVoyceb[n-2,x]}]

FullSimplify[Subtract@@{MorganVoyceB[n,x],(x+2)MorganVoyceB[n-1,x]-MorganVoyceB[n-2,x]}]

FullSimplify[Subtract@@{MorganVoyceb[n+1,x]MorganVoyceb[n-1,x]-MorganVoyceb[n,x]^2,x}]

FullSimplify[Subtract@@{MorganVoyceB[n+1,x]MorganVoyceB[n-1,x]-MorganVoyceB[n,x]^2,-1}]

DiffyQ

x(x+4)y''[x]+2(x+1)y'[x]-n(n+1)y[x]/.y->Function[x,Evaluate[MorganVoyceb[n,x]]]//FullSimplify

DSolve[x(x+4)y''[x]+2(x+1)y'[x]-n(n+1)y[x]==0,y[x],x]