The paper.  The polynomials Ban(p)(x) are studied in the paper  "On a novel iterative method to compute polynomial approximations to Bessel functions of the first kind and its connection to the solution of fractional diffusion/diffusion-wave problems" by S. Bravo Yuste and E. Abad   J. Phys. A: Math. Theor. 44 (2011) 075203.  Ban(p)(x)  is a polynomial of degree 2n with an independent coefficient equal to 1. 


The name. To call Badajoz polynomials to these functions makes sense because  Ba is the abbreviation of the name of the city and province of Badajoz ˗˗ the city where E. Abad and  I live. To call them Bessel approximation polynomials or even Bravo-Abad polynomials, could be suitable, but I think that "Badajoz polynomials" is a nicer name.


Two key properties of the Badajoz polynomials 


The normalized Bessel function. The relation between Bessel functions Jp(x) and Badajoz polynomials Ban(p)(x) can be better appreciated by means of the normalized Bessel function  :

                     

  zp being the first zero of Jp(x).

Bessel functions of the first kind for p=0,1,2

Normalized Bessel functions for p=0,1,2


Ban(p)(x) is an approximation to the normalized Bessel function. The polynomials Ban(p)(x)  are polynomial approximations to the normalized Bessel function , and consequently , also polynomial approximations to the Bessel function Jp(x):

                ↔            

This approximation improves when n increases:

       ↔          

The Badajoz function  Jp,n(x) defined above is then a polynomial approximation to the Bessel function of the first kind Jp(x).  This Badajoz function can be recursively calculated by means of this iterative formula:

with 

     ,

    ,   

  ,  


The generating integral operators   and  . Families of increasingly better approximation functions to  can be obtained by means of the repeated application of the integral operator     over a seed  function  f0(x). This operator is 

where

When the seed function is equal to one,  f0(x)=1, we get the family of Badajoz polynomials: 

For other seed functions f0(x), we get other families of approximations.  For example, for   f0(x)=1-x  we get  the  polynomials Ben(p)(x) (see S. Bravo Yuste and E. Abad J. Phys. A: Math. Theor. 44 (2011) 075203).


Integral operators   and with Mathematica:

f0[x_]:=1
Λ[0,p_]:=f0[x]
Λ[n_,p_]:=zp^2*Integrate[(1/u^(2p+1))*Integrate[x^(2p+1)*Λ[n-1,p],{x,0,u},Assumptions->Re[p]>-1&&Re[u]>0&&Re[zp]>0],{u, xx,1},Assumptions->Element[xx,Reals]&&Re[xx]>=0&&Re[zp]>0]/.xx->x
Λhat[n_,p_]:=(Lan=Λ[n,p];Lad=Lan/.x->0;Collect[Lan/Lad//Simplify,x])

These instructions generate Ban(p)(x). For example, Λhat[2,p] generates Ba2(p)(x).  One can get other families of approximations using other definitions for f0[x_]. For example, f0[x_]:=1-x leads to the Ben(p)(x) family.


Polynomials Ban(p)(x) with Mathematica.   An easy way to generate Ban(p)(x)  is by means of the following Mathematica instructions:

bag[x_,p_,0]:=1 ;
bag[x_,p_,m_]:=bag[x,p,m]=(-1)^m*p!*x^(2m)/(2^(2m) m! (m+p)!)-Sum[(-1)^k*p!/(2^(2k) k! (k+p)!)*bag[x,p,m-k],{k,1,m}]
Ba[x_,p_,n_]:=bag[x,p,n]/bag[0,p,n]

The Mathematica function  Ba[x_,p_,n_]  provides the polynomial Ban(p)(x).


Ban(0)(x)

The first 20 polynomials Ban(0)(x):

Plot of  Ban(0)(x).   Plot of the normalized Bessel function (dashed line) and the first 21 polynomials Ban(0)(x) with n=0,1,2,...20, (solid lines):

Plot of  J0,n(x).   Plot of the Bessel function J0(x) (dashed line) and the first 21 Badajoz functions J0,n(x) with  n=0,1,2,...20, (solid lines):


Ban(3/2)(x)

The first 11 polynomials Ban(3/2)(x):

Plot of  Ban(3/2)(x).  Plot of the normalized Bessel function  (dashed line) and the first 11 polynomials Ban(3/2)(x) with  n=0,1,2,...10,  (solid lines):

Plot of  J3/2,n(x).   Plot of the  Bessel function J3/2(x) (dashed line) and the first 21 Badajoz functions J3/2,n(x) with n=0,1,2,...20, (solid lines):


Diffusion and subdiffusion modes
 

Normal diffusion

Subdiffusion

d-dimensional equation

Boundary condition

c(R,t)=0

Initial condition c(r,0)=c0
Solution as superposition of modes

Temporal evolution of the modes

Spatial form of the modes

Plot of the first mode (j=1) for the 3D problem

Plot of the second mode (j=2) for the 3D problem