►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
One of the key properties of these polynomials is that they can be used to get increasingly better polynomial approximations to the Bessel function of the first kind Jp(x) (see below).
Besides, they describe the long-time decay modes of certain fractional diffusion and diffusion-wave problems (see below) just in the same way than Bessel functions Jp(x) describe normal-diffusion and wave modes of certain systems (see this link, for example).
►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 functionJp,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).
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:
for p=0,1,2
for p=0,1,2
↔ 
↔
,
,
,
and 
. Families of increasingly better approximation functions to 
(dashed line) and the first 21 polynomials Ban(0)(x) with n=0,1,2,…20, (solid lines):
(dashed line) and the first 11 polynomials Ban(3/2)(x) with n=0,1,2,…10, (solid lines):
