ABSTRACT (original article): In this paper we tackle the generalised Krawtchouk polynomials. We show a straightforward connection between certain auxiliary quantities involving the recurrence coefficients of these polynomials and the fifth Painlevé equation via iterated regularisation. This connection follows an algorithmic procedure avoiding the use of less standardised methods. We further explore how to obtain polynomial systems via iterated regularisation and find decomposition of certain birational transformations into the product of simple ones. CITATION (original article): Filipuk, G., Mañas–Mañas, J.F., Moreno–Balcázar, J.J. et al (2026). Differential system related to generalised Krawtchouk polynomials: iterated regularisation and Painlevé equation. Numer Algor 102, 1891–1922. https://doi.org/10.1007/s11075-025-02299-z
Connecting differential systems arising from recurrence coefficients of semiclassical orthogonal polynomials to the Painlevé equations usually requires finding complex birational transformations. Using Mathematica 14.1, we demonstrated how iterated regularisation on can systematically resolve points of indeterminacy of the differential system via blow-ups. Composing the chart transformations, we reduce the resulting final chart regular systems to a second-order scalar differential equation and explicitly identify it with the fifth Painlevé equation.
1
P
1
xP
A (simplified) implementation of the algorithm in the Wolfram Language is as follows.
The fifth Painlevé equation ) is given by
(
P
V
In[]:=
P5[x_,t_,α_,β_,γ_,δ_]:=D[x,{t,2}]-+-D[x,t]t+(αx+β/x)+γxt+δx==0
2
D[x,t]
1
2x
1
x-1
2
(x-1)
2
t
(x+1)
x-1
In[]:=
P5[y[t],t,α,β,γ,δ]
Out[]=
---+αy[t]+[t]-+[t]+[t]0
γy[t]
t
δy[t](1+y[t])
-1+y[t]
2
(-1+y[t])
β
y[t]
2
t
′
y
t
1
-1+y[t]
1
2y[t]
2
′
y
′′
y
Painlevé transcendents are non-linear special functions. At present they are not implemented in Mathematica. They possess many remarkable properties. For instance, solutions are meromorphic functions on the complex plane minus fixed singularities. Equation ) possesses non-trivial transformations, called Backlund transformations, which allow one to generate solutions for new values of the parameters α, β, γ, δ from a seed solution. One can find a lot of properties of the Painlevé equations at https://dlmf.nist.gov/32.
(
P
V
The system that we would like to reduce to is given by
()
P
v
In[]:=
syst=[t](mn+q[t]+2mq[t]+q[t]-αq[t]-mαq[t]+2mnp[t]q[t]-q[t]-2m-2+mn+mα-2p[t]),[t](p[t]+2mp[t]+p[t]-αp[t]-mαp[t]+2m+2+mt-mα++2mtp[t]q[t]+mt)
′
q
1
mt(p[t]+q[t])
2
p[t]
2
m
2
m
2
p[t]
2
q[t]
2
m
2
q[t]
2
q[t]
2
q[t]
2
m
2
q[t]
′
p
1
mt(p[t]+q[t])
2
m
2
p[t]
2
m
2
p[t]
2
p[t]
2
p[t]
2
m
3
p[t]
2
q[t]
Out[]=
[t](mn+q[t]+2mq[t]+q[t]-αq[t]-mαq[t]+2mnp[t]q[t]-q[t]-2m-2+mn+mα-2p[t]),[t]
′
q
1
mt(p[t]+q[t])
2
p[t]
2
m
2
m
2
p[t]
2
q[t]
2
m
2
q[t]
2
q[t]
2
q[t]
2
m
2
q[t]
′
p
p[t]+2mp[t]+p[t]-αp[t]-mαp[t]+2m+2+mt-mα++2mtp[t]q[t]+mt
2
m
2
p[t]
2
m
2
p[t]
2
p[t]
2
p[t]
2
m
3
p[t]
2
q[t]
mt(p[t]+q[t])
Here and are some parameters, which will be later on connected to the parameters of ). As we can easily check, we cannot derive a scalar second order first degree equation for the functions and :
m
n
(
P
V
q
p
In[]:=
Solve[syst[[1]],p[t]]
Out[]=
p[t]2mnq[t]-2-mt[t]-(-4(-mn+q[t])(-q[t]-2mq[t]-q[t]+αq[t]+mαq[t]+2m+2-mn-mα+mtq[t][t])),p[t]2mnq[t]-2-mt[t]+(-4(-mn+q[t])(-q[t]-2mq[t]-q[t]+αq[t]+mαq[t]+2m+2-mn-mα+mtq[t][t]))
1
2(-mn+q[t])
2
m
2
m
2
q[t]
′
q
2
(-2mnq[t]+2+mt[t])
2
m
2
q[t]
′
q
2
m
2
m
2
q[t]
2
m
2
q[t]
2
q[t]
2
q[t]
′
q
1
2(-mn+q[t])
2
m
2
m
2
q[t]
′
q
2
(-2mnq[t]+2+mt[t])
2
m
2
q[t]
′
q
2
m
2
m
2
q[t]
2
m
2
q[t]
2
q[t]
2
q[t]
′
q
In[]:=
Solve[syst[[1]],q[t]]
Out[]=
q[t]1+2m+-α-mα+2mnp[t]--mt[t]--4(2m+2-mn-mα+2p[t])(-mn+mtp[t][t])(2(2m+2-mn-mα+2p[t])),q[t]1+2m+-α-mα+2mnp[t]--mt[t]+-4(2m+2-mn-mα+2p[t])(-mn+mtp[t][t])(2(2m+2-mn-mα+2p[t]))
2
m
2
m
2
p[t]
′
q
2
(-1-2m-+α+mα-2mnp[t]++mt[t])
2
m
2
m
2
p[t]
′
q
2
m
2
m
2
p[t]
′
q
2
m
2
m
2
m
2
m
2
p[t]
′
q
2
(-1-2m-+α+mα-2mnp[t]++mt[t])
2
m
2
m
2
p[t]
′
q
2
m
2
m
2
p[t]
′
q
2
m
2
m
Square roots suggest that both functions and satisfy a second order second degree ordinary differential equation and, hence, connection to the fifth Painlevé equation is very non-trivial.
q
p
We shall start examining the system on the charts of and find, if any, points of indeterminacy, where the right-hand side in at least one of the equations becomes 0/0.
1
P
1
xP
1. Chart (q,p):
In[]:=
syst[[1]][[2]]//NumeratorDenominator
Out[]=
{mn+q[t]+2mq[t]+q[t]-αq[t]-mαq[t]+2mnp[t]q[t]-q[t]-2m-2+mn+mα-2p[t],mt(p[t]+q[t])}
2
p[t]
2
m
2
m
2
p[t]
2
q[t]
2
m
2
q[t]
2
q[t]
2
q[t]
2
m
2
q[t]
In[]:=
Solve[%==0,{q[t],p[t]}]
Out[]=
{q[t]0,p[t]0},q[t],p[t],q[t],p[t]
1+m
m
-1-m
m
1+m-α
m
-1-m+α
m
In[]:=
syst[[2]][[2]]//NumeratorDenominator
Out[]=
{p[t]+2mp[t]+p[t]-αp[t]-mαp[t]+2m+2+mt-mα++2mtp[t]q[t]+mt,mt(p[t]+q[t])}
2
m
2
p[t]
2
m
2
p[t]
2
p[t]
2
p[t]
2
m
3
p[t]
2
q[t]
In[]:=
Solve[%==0,{q[t],p[t]}]
Out[]=
{q[t]0,p[t]0},q[t],p[t],q[t],p[t]
1+m
m
-1-m
m
1+m-α
m
-1-m+α
m
As we can see, there are several points of indeterminacy.
2. Chart (q,1/p):
In[]:=
syst/.p[t]->1/P[t]/.D[p[t]->1/P[t],t]
Out[]=
[t]+q[t]+2mq[t]+q[t]-αq[t]-mαq[t]-q[t]+-2m-2+mn+mα-,-[t]+++-+++--++mt
′
q
mn
2
P[t]
2
m
2
m
2
P[t]
2mnq[t]
P[t]
2
q[t]
2
m
2
q[t]
2
q[t]
2
q[t]
2
2
m
2
q[t]
P[t]
mt+q[t]
1
P[t]
′
P
2
P[t]
2
m
3
P[t]
2m
2
P[t]
2
2
m
2
P[t]
mt
2
P[t]
mα
2
P[t]
1
P[t]
2m
P[t]
2
m
P[t]
α
P[t]
mα
P[t]
2mtq[t]
P[t]
2
q[t]
mt+q[t]
1
P[t]
In[]:=
systqP=Solve[%,{q'[t],P'[t]}]//Flatten//Simplify
Out[]=
[t],[t]-+m(2+2m+t-α)P[t]+mt+((1+m)(1+m-α)+2mtq[t])
′
q
mn+(-+2mnP[t]+(1+m)(1+m-α))q[t]+mP[t](-2m+(-2-2m+n+α)P[t])
2
m
2
P[t]
2
q[t]
mtP[t](1+P[t]q[t])
′
P
2
m
3
P[t]
2
q[t]
2
P[t]
mt(1+P[t]q[t])
In[]:=
systqP[[1]][[2]]//NumeratorDenominator
Out[]=
{mn+(-+2mnP[t]+(1+m)(1+m-α))q[t]+mP[t](-2m+(-2-2m+n+α)P[t]),mtP[t](1+P[t]q[t])}
2
m
2
P[t]
2
q[t]
In[]:=
Solve[%==0,{q[t],P[t]}]//Simplify
Out[]=
q[t],P[t]0,q[t]1+,P[t]-,q[t],P[t]-
n
m
1
m
m
1+m
1+m-α
m
m
1+m-α
In[]:=
syst[[2]][[2]]//NumeratorDenominator
Out[]=
{p[t]+2mp[t]+p[t]-αp[t]-mαp[t]+2m+2+mt-mα++2mtp[t]q[t]+mt,mt(p[t]+q[t])}
2
m
2
p[t]
2
m
2
p[t]
2
p[t]
2
p[t]
2
m
3
p[t]
2
q[t]
In[]:=
Solve[%==0,{q[t],P[t]}]//Simplify
Out[]=
{}
As we can see, some points are the same as in the -chart but there is also a new point which did not appear before, namely, .
(q,p)
q[t],P[t]0
n
m
3. Chart (1/q,p):
In[]:=
syst/.q[t]->1/Q[t]/.D[q[t]->1/Q[t],t]
Out[]=
-[t],[t]
′
Q
2
Q[t]
mn--++-+++--+-
2
p[t]
2m
2
Q[t]
2
2
m
2
Q[t]
mn
2
Q[t]
mα
2
Q[t]
2p[t]
2
m
2
Q[t]
1
Q[t]
2m
Q[t]
2
m
Q[t]
α
Q[t]
mα
Q[t]
2mnp[t]
Q[t]
2
m
2
p[t]
Q[t]
mtp[t]+
1
Q[t]
′
p
p[t]+2mp[t]+p[t]-αp[t]-mαp[t]+2m+2+mt-mα+++
2
m
2
p[t]
2
m
2
p[t]
2
p[t]
2
p[t]
2
m
3
p[t]
mt
2
Q[t]
2mtp[t]
Q[t]
mtp[t]+
1
Q[t]
In[]:=
systQp=Solve[%,{Q'[t],p'[t]}]//Flatten//Simplify
Out[]=
[t]-,[t]
′
Q
Q[t](m(-2-2m+n+α)+(1+m)(1+m-α)Q[t]-2mp[t](m-nQ[t])+mQ[t](-m+nQ[t]))
2
p[t]
mt(1+p[t]Q[t])
′
p
mt+m(2+2m+t-α)++p[t]Q[t](2mt+(1+m)(1+m-α)Q[t])
2
p[t]
2
Q[t]
2
m
3
p[t]
2
Q[t]
mtQ[t](1+p[t]Q[t])
In[]:=
systQp[[1]][[2]]//NumeratorDenominator
Out[]=
{-Q[t](m(-2-2m+n+α)+(1+m)(1+m-α)Q[t]-2mp[t](m-nQ[t])+mQ[t](-m+nQ[t])),mt(1+p[t]Q[t])}
2
p[t]
In[]:=
Solve[%==0,{Q[t],p[t]}]//Simplify
Out[]=
Q[t],p[t]-,Q[t],p[t]
m
1+m
1+m
m
m
1+m-α
-1-m+α
m
In[]:=
syst[[2]][[2]]//NumeratorDenominator
Out[]=
{p[t]+2mp[t]+p[t]-αp[t]-mαp[t]+2m+2+mt-mα++2mtp[t]q[t]+mt,mt(p[t]+q[t])}
2
m
2
p[t]
2
m
2
p[t]
2
p[t]
2
p[t]
2
m
3
p[t]
2
q[t]
In[]:=
Solve[%==0,{Q[t],p[t]}]//Simplify
Out[]=
{}
4. Chart (1/q,1/p):
In[]:=
syst/.q[t]->1/Q[t]/.D[q[t]->1/Q[t],t]/.p[t]->1/P[t]/.D[p[t]->1/P[t],t]
Out[]=
-[t]--++-+++---Q[t]+,-[t]+++-+++--++
′
Q
2
Q[t]
mn
2
P[t]
2m
2
Q[t]
2
2
m
2
Q[t]
mn
2
Q[t]
mα
2
Q[t]
2
2
m
P[t]
2
Q[t]
1
Q[t]
2m
Q[t]
2
m
Q[t]
α
Q[t]
mα
Q[t]
2
m
2
P[t]
2mn
P[t]Q[t]
mt+
1
P[t]
1
Q[t]
′
P
2
P[t]
2
m
3
P[t]
2m
2
P[t]
2
2
m
2
P[t]
mt
2
P[t]
mα
2
P[t]
1
P[t]
2m
P[t]
2
m
P[t]
α
P[t]
mα
P[t]
mt
2
Q[t]
2mt
P[t]Q[t]
mt+
1
P[t]
1
Q[t]
In[]:=
systQP=Solve[%,{Q'[t],P'[t]}]//Flatten//Simplify
Out[]=
[t]-,[t]-
′
Q
Q[t](-2mP[t](m-nQ[t])+mQ[t](-m+nQ[t])+(m(-2-2m+n+α)+(1+m)(1+m-α)Q[t]))
2
P[t]
mtP[t](P[t]+Q[t])
′
P
mt++m(2+2m+t-α)P[t]+Q[t](2mt+(1+m)(1+m-α)Q[t])
3
P[t]
2
m
2
Q[t]
2
Q[t]
2
P[t]
mtQ[t](P[t]+Q[t])
In[]:=
systQP[[1]][[2]]//NumeratorDenominator
Out[]=
{-Q[t](-2mP[t](m-nQ[t])+mQ[t](-m+nQ[t])+(m(-2-2m+n+α)+(1+m)(1+m-α)Q[t])),mtP[t](P[t]+Q[t])}
2
P[t]
Summing up, we can now identify different points, which we will blow up next.
Let us blow up the first point p[1].
We can check that the right-hand side is regular on the exceptional divisor:
However, the right-hand side of the second equation is not polynomial and we can easily check that there are other points of indeterminacy which are not on the exceptional divisor:
Hence, we can restore the Hamiltonian function. However, we do not need it. Let us look at the changes of variables that we made:
We can recalculate the 2-form
Hence, the original system will also be Hamiltonian with the non-standard 2-form. We can easily restore the Hamiltonian function
Hence, the Hamiltonian is given by
This can be easily verified:
Now let us examine the relation of systuviter to the fifth Painlevé equation.
We obtain an algebraic system for the unknown coefficients. With additional conditions it has solutions:
Choosing, for instance, the second one,
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Differential system related to generalised Krawtchouk polynomials: iterated regularisation and Painlevé equation
by Galina Filipuk, Juan F. Mañas-Mañas, Juan J. Moreno-Balcázar, Cristina Rodríguez-Perales
Wolfram Community, STAFF PICKS, October 16, 2026
https://community.wolfram.com/t/28352
by Galina Filipuk, Juan F. Mañas-Mañas, Juan J. Moreno-Balcázar, Cristina Rodríguez-Perales
Wolfram Community, STAFF PICKS, October 16, 2026
https://community.wolfram.com/t/28352