Determining connection coefficients with parameter initial conditions.

*Requirements: ore_algebra package*

In [1]:

```
# Define the rings of differential and shift operators
from ore_algebra import *
Ind.<n> = PolynomialRing(QQ); Shift.<Sn> = OreAlgebra(Ind)
Pols.<z> = PolynomialRing(QQ); Diff.<Dz> = OreAlgebra(Pols)
```

In [2]:

```
# Start with the differential equation
diff = 5*z^2*(2*z-1)*(z-3)*Dz^4 + 2*z*(59*z^2-139*z+36)*Dz^3 + 6*(61*z^2-80*z+6)*Dz^2 + 12*(25*z-11)*Dz + 36
```

In [3]:

```
# Get recurrence on coefficients
rec = diff.to_S(Shift).primitive_part()
rec
```

Out[3]:

(15*n^2 + 57*n + 36)*Sn^2 + (-35*n^2 - 103*n - 66)*Sn + 10*n^2 + 28*n + 18

In [4]:

```
# The following gives asymptotic expansions for a basis of solutions to the recurrence
show(rec.generalized_series_solutions(n=3))
```

In [5]:

```
# There is a basis with the following series expansions near the origin
print(diff.local_basis_expansions(0, order = 4))
```

[z^(-1), 1 - 1/2*z^2, z + 11/6*z^2, z^(6/5) + 11/6*z^(11/5) + 187/63*z^(16/5) + 1705/351*z^(21/5)]

In [6]:

```
# A power series solution starting f0 + f1*z + ... is expressed in this basis with coordinates
var('f0 f1')
ini = [0,f0,f1,0]
ini
```

Out[6]:

[0, f0, f1, 0]

In [7]:

```
# This generating function potentially has singularities at z = 1/2 and z = 3
diff.leading_coefficient().factor()
```

Out[7]:

(10) * (z - 3) * (z - 1/2) * z^2

In [8]:

```
# Expansions of a basis near z=1/2
for k in diff.local_basis_expansions(1/2, order = 4):
show(k)
```

In [9]:

```
# We can compute the change of basis matrix between the basis with series at the origin and the basis with series at 1/2
# We take a low numeric accuracy here to fit on screen
show(diff.numerical_transition_matrix([0,1/2],1e-1))
```

In [10]:

```
# The coefficient of the dominant singular term is f0/2 - 3f1/2
bas = diff.numerical_transition_matrix([0,1/2],1e-10) * vector(ini)
show(bas[0])
```

In [11]:

```
# Expansions of a basis near z=3
for k in diff.local_basis_expansions(3, order = 5):
show(k)
```

In [12]:

```
# We can compute the change of basis matrix between the basis with series at the origin and the basis with series at 3
# We take a low numeric accuracy here to fit on screen
# Note -- the path for continuation must move off the real line to *avoid* the singularity at z=1/2
show(diff.numerical_transition_matrix([0,I,3],1e-1))
```

In [13]:

```
# The coefficient of the singular term is (9/2)(f1-f0)
bas2 = diff.numerical_transition_matrix([0,I,3],1e-10) * vector(ini)
show(bas2[0])
```

In [ ]:

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