Computing Hypergeometric Solutions of Second Order Linear Differential Equations using Quotients of Formal Solutions and Integral Bases

1 Introduction

A second order homogeneous linear differential equation with rational function coefficients $A_{i} \in Q (x)$

A_{2} y^{''} + A_{1} y^{'} + A_{0} y = 0

(3)

corresponds to the differential operator

L = A_{2} \partial^{2} + A_{1} \partial + A_{0} \in Q (x) [\partial]

where $\partial = \frac{d}{d x}$ . Another representation of (3) is

L (y) = 0.

This paper gives two heuristic (see Remarks

1.2 and 3.4) algorithms to find a hypergeometric solution of (3) in the form (1) and (2). The form (2) is more general than in prior works Fang and van Hoeij (2011); Kunwar and van Hoeij (2013); van Hoeij and Vidunas (2015); Kunwar (2014). Papers van Hoeij and Vidunas (2015) and Kunwar (2014) were restricted to a specific number of singularities (4 in van Hoeij and Vidunas (2015) and 5 in Kunwar (2014)). Papers Kunwar and van Hoeij (2013) and Fang and van Hoeij (2011) were restricted to specific degrees (degree 3 in Kunwar and van Hoeij (2013) and a degree-2 decomposition in Fang and van Hoeij (2011)). Our algorithms are not restricted to a specific number of singularities or a specific degree. Moreover, our algorithms can find algebraic functions

f

in (1) and (2).

Our first algorithm, Algorithm 3.1, tries to find solutions of (3) in the form of (1). Our second algorithm, Algorithm 4.5, tries to reduce equations with solutions of form (2) to equations and then calls our first algorithm.

We assume that (3) has no Liouvillian solutions and hence irreducible. Otherwise one can solve (3) with Kovacic’s algorithm Kovacic (1986).

Let $L_{i n p}$ $\in$ $Q (x) [\partial]$ ( $L$ input) be a second order linear differential operator, regular singular (details in Section 2.1) and without Liouvillian solutions:

Goal 1: (Algorithm 3.1) Find a solution of form (1) if it exists.
Goal 2: (Algorithm 4.5) Try to transform $L_{i n p}$ to a simpler operator (which hopefully has a solution in form (1)).

The crucial steps for Goal 1 are to find (candidates for) $a_{1}$ , $a_{2}$ , $b_{1}$ and the pullback function $f$ . Finding the parameters $a_{1}$ , $a_{2}$ , $b_{1}$ is the combinatorial part; Theorem 3.2 helps us to eliminate the vast majority of cases. Given $a_{1}, a_{2}, b_{1}$ (or equivalently a base operator $L_{B}$ ), if we know the value of a certain constant $c$ , by comparing quotients of formal solutions of $L_{B}$ and $L_{i n p}$ , we can compute $f$ . We have no direct formula for $c$ ; to obtain it with a finite computation, we take a prime number $ℓ$ . Then, for each $c \in {1, \dots, ℓ - 1}$ we try to compute $f$ modulo $ℓ$ . If this succeeds, then we lift $f$ modulo a power of $ℓ$ , and try rational number reconstruction.

Goal 2 is to find a transformation to convert $L_{i n p}$ to a simpler operator ${~ L}_{i n p}$ . The key idea is to follow the strategy of the POLRED algorithm in Cohen and Diaz Y Diaz (1991). It takes as input a polynomial $L_{i n p} \in Q [x]$ and finds an element $G \in Q [x] / Q [x] L_{i n p}$ whose minimal polynomial is close to optimal. It works as follows:

“Finite points”: Compute an integral basis.
“Valuations at infinity”: Find an integral element $G$ with (near) optimal absolute values.

Our key idea is to apply POLRED’s strategy to $L_{i n p} \in Q (x) [\partial]$ . First compute an integral basis (introduced in Kauers and Koutschan (2015)). Then normalize at infinity (following (Trager, 1984, Section 2.3) where this is done for function fields) so we can select an element with minimal valuations at infinity.

Example 1.1 (Rational Pullback Function).

The differential operator

L_{i n p} = 147 x (x - 1) (x + 1) \partial^{2} + (266 x^{2} - 42 x - 98) \partial + 20 x - 5

has a $_{2} F_{1}$ -type solution in the form of (1), which is

Y (x) = exp (\int r d x) \cdot_{2} F_{1} (\frac{5}{42}, \frac{11}{42}; \frac{2}{3}; f)

where

exp (\int r d x) = {(x + 1)}^{- \frac{5}{21}} a n d f = \frac{4 x}{{(x + 1)}^{2}}

(4)

Section 3.4 shows how to find the parameters $a_{1}, a_{2}, b_{1}$ $=$ $\frac{5}{42}, \frac{11}{42}, \frac{2}{3}$ . Then $f$ is computed with the quotient method:

Remark 1.1 (The Quotient Method).

The hypergeometric function

_{2} F_{1} (\frac{5}{42}, \frac{11}{42}; \frac{2}{3}; x)

is a solution of the Gauss hypergeometric differential operator

L_{B} = \partial^{2} + \frac{(29 x - 14)}{21 x (x - 1)} \partial + \frac{55}{1764 x (x - 1)} .

$L_{B}$ has two solutions at $x = 0$ . They are

	$y_{1} (x)$	$=_{2} F_{1} (\frac{5}{41}, \frac{11}{42}; \frac{2}{3}, x) = 1 + \frac{55}{1176} x + \dots,$
	$y_{2} (x)$	$= x^{\frac{1}{3}} (1 + \frac{475}{2352} x + \frac{1941325}{19361664} x^{2} + \dots) .$

The so-called exponents of $L_{B}$ at $x = 0$ are the exponents of $x$ in the dominant terms of the solutions $y_{1}$ and $y_{2}$ , so the exponents are $e_{0, 1} = 0$ and $e_{0, 2} = \frac{1}{3}$ . The minimal operator for $y (f)$ has the following solutions at $x = 0$ :

	$y_{1} (f)$	$= 1 + \frac{55}{294} x - \frac{4939}{86436} x^{2} + \frac{16135823}{} 304946208 x^{3} + \dots,$
	$y_{2} (f)$	$= c \cdot x^{\frac{1}{3}} (1 + \frac{83}{588} x + \frac{6805}{1210104} x^{2} + \dots)$

for some constant $c$ that depends on $f$ . The exponents are again $0$ and $\frac{1}{3}$ , because $x = 0$ is a root of $f$ with multiplicity $e_{0} = 1$ (Theorem 2.1). Let

	$Y_{1} (x)$	$= exp (\int r d x) y_{1} (f) = 1 - \frac{5}{98} x + \frac{439}{9604} x^{2} + \dots,$		(5)
	$Y_{2} (x)$	$= exp (\int r d x) y_{2} (f) = c \cdot x^{\frac{1}{3}} (1 - \frac{19}{196} x + \dots) .$		(6)

(5) and (6) form a basis of solutions of $L_{i n p}$ . Here $exp (\int r d x)$ is the same as in (4). Denote the quotients of the formal solutions of $L_{B}$ and $L_{i n p}$ by

	$q$	$= \frac{y_{1} (x)}{y_{2} (x)}$
	$Q$	$= \frac{Y_{1} (x)}{Y_{2} (x)} = \frac{y_{1} (f)}{y_{2} (f)} = q (f)$

respectively. It follows that $q^{- 1} (Q (x))$ gives a series expansion of $f$ at $x = 0$ . Given enough terms we can compute $f$ with rational function reconstruction. This Quotient Method was already used in (van Hoeij and Vidunas, 2015, Section 5.1). In order to turn this into an algorithm for solving differential equations we need to answer the following questions:

[label=Q0.]
How many terms are needed to reconstruct $f$ ? This is equivalent to finding a degree bound for $f$ .
How to find the parameters $a_{1}$ , $a_{2}$ , $b_{1}$ ? This is the combinatorial part of our algorithm.
The exponents $0$ , $\frac{1}{3}$ of $L_{i n p}$ at $x = 0$ only determine $\frac{Y_{1}}{Y_{2}}$ up to a constant factor (see Remark 2.2 in Section 2.3). This means the quotient $\frac{y_{1} (f)}{y_{2} (f)}$ is only known up to a constant $c$ . How to find this constant?
What if $L_{i n p}$ has logarithmic solutions at $x = 0$ ?
What if $f$ is an algebraic function?
What if $L_{i n p}$ does not have solutions in the form of (1), but has solutions in the form of (2)?

Remark 3.2, Section 3.4, Section 3.6, and Section 3.5.2 provide answers to Q1, Q2, Q3, and Q4 respectively. Section 3.6 answers Q5. The parts Q1,…,Q5 are already in our ISSAC 2015 paper Imamoglu and van Hoeij (2015). Algorithm 4.5, which finds solutions of form (2), is new compared to Imamoglu and van Hoeij (2015). So Q6 is the main new part in this paper. It will be discussed in Section 4. Example 1.3 will illustrate to Q6.

Remark 1.2.

Both algorithms are very effective in practice but they are not proven. For completeness for Goal 1 we still need a theorem for good prime numbers. A good prime is a prime for which reconstruction will work.

Example 1.2 (Algebraic Pullback Function).

The differential operator

L_{i n p} = \partial^{2} + \frac{1}{4} \frac{x^{4} - 44 x^{3} + 1206 x^{2} - 44 x + 1}{{(x^{2} - 34 x + 1)}^{2} x^{2}}

has a $_{2} F_{1}$ -type solution in the form of (1), which is

Y (x) = exp (- \frac{1}{2} \int r d x) \cdot_{2} F_{1} (\frac{1}{3}, \frac{2}{3}; 1; f)

where $r =$

\frac{- x^{5} + 22 x^{4} - 55 x^{3} - 343 x^{2} + 58 x - 1 + 6 x (x^{2} - 7 x + 1) \sqrt{x^{2} - 34 x + 1}}{x (x^{4} - 41 x^{3} + 240 x^{2} - 41 x + 1) (x + 1)}

and

f = \frac{1}{2} \frac{1 + 30 x - 24 x^{2} + x^{3} - (x^{2} - 7 x + 1) \sqrt{x^{2} - 34 x + 1}}{1 + 3 x + 3 x^{2} + x^{3}} .

Here the pullback function $f$ is an algebraic function: $Q (x, f)$ is an algebraic extension of $Q (x)$ of degree $a_{f} = 2$ ( $a_{f}$ is $1$ if and only if $f$ is a rational function, as in Example 1.1). Algorithm 3.1 can find this solution.

Example 1.3 (Finding Solutions in the form of (2) using an Integral Basis).

Consider the differential operator²²2Prof. Jean-Marie Maillard sent us this differential operator.

	$L_{i n p} = \partial^{2}$	$- \frac{512 x^{5} + 384 x^{4} - 64 x^{3} - 88 x^{2} - 10 x - 1}{x (4 x - 1) (4 x + 1) (16 x^{3} + 24 x^{2} + 5 x + 1)} \partial$
		$+ \frac{512 x^{5} + 64 x^{4} - 128 x^{3} - 60 x^{2} - 8 x - 1}{} x^{2} (4 x - 1) (4 x + 1) (16 x^{3} + 24 x^{2} + 5 x + 1) .$

Algorithm 3.1 can not solve $L_{i n p}$ . We try to transform $L_{i n p}$ to simpler operator ${~ L}_{i n p}$ . First we compute an integral basis. Then we normalize the basis at infinity and obtain $[B_{0}, B_{1}]$ where

	$B_{0} =$	$\frac{16 x^{4} - x^{2}}{(16 x^{3} + 24 x^{2} + 5 x + 1) x} \partial$
		$+ \frac{- 34359738400 x^{3} - 51539607556 x^{2} - 10737418241 x - 2147483648}{(16 x^{3} + 24 x^{2} + 5 x + 1) x}$

and

B_{1} = \frac{16 x^{3} - x}{(16 x^{3} + 24 x^{2} + 5 x + 1) x} \partial + \frac{- 32 x^{2} - 4 x - 1}{(16 x^{3} + 24 x^{2} + 5 x + 1) x} .

We try to find a suitable $G \in Q (x) [\partial] / Q (x) [\partial] L_{i n p}$ . It should be a combination of $B_{0}$ and $B_{1}$ . For this example, we take $G = B_{1}$ . This $G$ is called a gauge transformation. It maps solutions of $L_{i n p}$ to solutions of

{~ L}_{i n p} = \partial^{2}

+ \frac{48 x^{2} - 1}{x (16 x^{2} - 1)} \partial + \frac{16}{16 x^{2} - 1} .

${~ L}_{i n p}$ has a solution in the form of (1),

y (x) =_{2} F_{1} (\frac{1}{2}, \frac{1}{2}; 1; 16 x^{2})

which is easy to find with Algorithm 3.1. Then we apply the inverse gauge transformation and obtain a solution of $L_{i n p}$ in the form of (2), which is

2 Preliminaries

This section recalls the concepts needed in later sections.

2.1 Differential Operators, Singularities, Formal Solutions

We start with some classical definitions which can be also found in Ince (1926); van der Put and Singer (2003); Kunwar (2014); Fang (2012); Debeerst (2007); Yuan (2012).

Definition 2.1.

Let $L = \sum_{i = 0}^{n} A_{i} \partial^{i} \in C (x) [\partial]$ be an operator of order $n$ .

[label=()]
A point $p \in C$ is called a singularity of $L$ if it is a zero of the leading coefficient of $L$ or a pole of any other coefficient of $L$ . The point $p = \infty$ is called a singularity if $p = 0$ is a singularity of $L_{1 / x}$ . Here $L_{1 / x}$ is the differential operator obtained from $L$ via a change of variables $x \mapsto \frac{1}{x}$ (note that $x \mapsto f$ sends $\partial$ to $\frac{1}{f^{'}} \partial)$ .
If $x = p$ is not a singularity, then it is called a regular point of $L$ .
A singularity $p \in C$ is called a regular singularity if $(x - p)^{i} \frac{A_{n - i}}{A_{n}}$ is analytic at $x = p$ for $1 \leq i \leq n - 1$ . The point $p = \infty$ is a regular singularity if $p = 0$ is a regular singularity of $L_{1 / x}$ .
$L$ is regular singular if all its singularities are regular singular.

2.2 Gauss Hypergeometric Differential Operator and $_{2} f_{1}$ Function

Definitions also can be found in Wang and Guo (1989); Kunwar (2014); Fang (2012). Let $a_{1}$ , $a_{2}$ , $b_{1}$ $\in$ $Q$ . The operator

L_{B} = x (1 - x) \partial^{2} + (b_{1} - (a_{1} + a_{2} + 1) x) \partial - a_{1} a_{2}

is called Gauss hypergeometric differential operator (GHDO). The solution space of $L_{B}$ in a universal extension Fang (2012) has dimension 2 because the order of $L_{B}$ is 2. One of the solutions of $L_{B}$ at $x = 0$ is the Gauss hypergeometric function. It is denoted by $_{2} F_{1}$ and defined by the Gauss hypergeometric series

_{2} F_{1} (a_{1}, a_{2}; b_{1}; x) = \infty \sum k = 0 \frac{(a_{1})_{k} (a_{2})_{k}}{(b_{1})_{k} k!} x^{k} .

Here $(λ)_{k}$ denotes the Pochammer symbol. It is defined as $(λ)_{k} = λ (λ + 1) \dots (λ + k - 1)$ and $(λ)_{0} = 1$ .

$L_{B}$ has three regular singularities at the points $x = 0$ , $x = 1$ , and $x = \infty$ , with exponents ${0, 1 - b_{1}}$ , ${0, b_{1} - a_{1} - a_{2}}$ , and ${a_{1}, a_{2}}$ respectively. We denote the exponent differences of a GHDO as $α_{0} = | 1 - b_{1} |$ , $α_{1} = | b_{1} - a_{1} - a_{2} |$ , $α_{\infty} = | a_{1} - a_{2} |$ . We may assume ${a_{1}, a_{2}, b_{1} - a_{1}, b_{1} - a_{2}} \cap Z = \emptyset$ , otherwise $L_{B}$ is reducible (it has exponential solutions).

Let $d_{i}$ be $\infty$ if $α_{i} \in Z$ , and the denominator of $α_{i}$ if $α_{i} \in Q - Z$ . We will only consider $a_{1}, a_{2}, b_{1}$ for which $L_{B}$ has no Liouvillian solutions. From the Schwarz list Schwarz (1873) one finds that this is equivalent to $\frac{1}{d_{0}} + \frac{1}{d_{1}} + \frac{1}{d_{\infty}} < 1$ .

2.3 Transformations and Singularities

We summarize properties of transformations in this section. These properties can also be found in Kunwar (2014); Fang (2012); Debeerst (2007); Yuan (2012).

Let $L_{1} \in C (x) [\partial]$ be a differential operator of order $2$ , and let $y$ be a solution of $L_{1}$ . We consider the following transformations that send solutions of $L_{1}$ to solutions of another second order differential operator $L_{2}$ .

Change of variables:
$y (x) ⟶ y (f)$ , where $f \in ¯ ¯¯¯¯¯¯¯¯¯¯ ¯ Q (x)$ .
For $L_{1}$ this means substituting $(x, \partial) \mapsto (f, \frac{1}{f^{'}} \partial)$ .
Notation: $L_{1} {f \to}_{C} L_{2}$ .
Exp-product:
$y (x) ⟶ exp (\int r d x) y (x)$ , where $r \in ¯ ¯¯¯¯¯¯¯¯¯¯ ¯ Q (x)$ .
For $L_{1}$ this means $\partial \mapsto \partial - r$ .
Notation: $L_{1} {r \to}_{E} L_{2}$ .
Gauge transformation:
$y (x) ⟶ r_{0} \cdot y (x) + r_{1} \cdot y (x)$ , where $r_{0}, r_{1} \in Q (x)$ .
For $L_{1}$ this means computing the least common left multiple of $L_{1}$ and $r_{1} \partial + r_{0}$ , and right-dividing it by $r_{1} \partial + r_{0}$ .
Notation: $L_{1} {r_{0}, r_{1} - -- \to}_{G} L_{2}$ .

Remark 2.1.

Transformations can affect singularities and exponents.

[label=()]
If a transformation ${r \to}_{E}$ can send a singular point $x = p$ to a regular point $x = p$ , then we call $x = p$ a false singularity.
A singularity $x = p$ is a false singularity Debeerst (2007) if and only if $x = p$ is not logarithmic and the exponent difference is 1.
If $x = p$ is a singularity of $L_{1}$ and if transformation ${r \to}_{E}$ can send $L_{1}$ to an equation $L_{2}$ for which all solutions of $L_{2}$ are analytic at $x = p$ , then we call $x = p$ a removable singularity.
A point $x = p$ is removable Debeerst (2007) if and only if $x = p$ is not logarithmic and the exponent difference is an integer. Non-removable singularities are called true singularities.
A point $x = p$ is a true singularity if and only if the exponent difference is not an integer or $x = p$ is logarithmic.
If $L_{1} {r_{0}, r_{1} - -- \to}_{G} L_{2}$ , then $L_{1}$ and $L_{2}$ are called gauge equivalent. If $V_{1}$ and $V_{2}$ are the solution spaces of $L_{1}$ and $L_{2}$ respectively, then $G = r_{1} \partial + r_{0}$ maps $V_{1}$ to $V_{2}$ , i.e., $G (V_{1}) = V_{2}$ .

Remark 2.2.

The quotient method (Remark 1.1 in Section 1) can only use true singularities, otherwise, $\frac{Y_{1}}{Y_{2}}$ , the quotients of solutions of $L_{i n p}$ , would only be known up to a Möbius transformation instead of up to a constant.

Remark 2.3.

At the moment, Algorithms

3.1 and 4.5 are only implemented for rational function coefficients. However, if

f, r, r_{0}, r_{1}

are algebraic, then the three transformations may turn an operator with rational function coefficients into an operator with algebraic function coefficients.

Theorem 2.1.

Bostan et al. (2011) Let the GHDO $L_{B}$ have exponent differences $α_{0}$ at $x = 0$ , $α_{1}$ at $x = 1$ , and $α_{\infty}$ at $x = \infty$ . Let $L_{B} {f \to}_{C} L_{i n p}$ . If $f (p) \in {0, 1, \infty}$ , then $L_{i n p}$ has the following exponent difference at $x = p$ :

$α_{0} \cdot e_{p}$ if $f$ has a zero at $x = p$ with multiplicity $e_{p}$ ,
$α_{1} \cdot e_{p}$ if $f - 1$ has a zero at $x = p$ with multiplicity $e_{p}$ ,
$α_{\infty} \cdot e_{p}$ if $f$ has a pole at $x = p$ with order $e_{p}$ .

3 Computing Solutions of a Second Order Linear Differential Operator in the form of (1) by using Quotients of Formal Solutions

This section gives our first algorithm, which looks for solutions of the form of (1).

3.1 Problem Statement

Given a second order linear differential operator $L_{i n p} \in Q (x) [\partial]$ , irreducible and regular singular, we want to find a $_{2} F_{1}$ -type solution of the differential equation $L_{i n p} (y) = 0$ of the form of (1). This is equivalent to finding transformations 1 and 2 from a GHDO $L_{B}$ to $L_{i n p}$ . Therefore, we need to find

$L_{B}$ (i.e., find $a_{1}, a_{2}, b_{1}$ ),
parameters $f$ and $r$ of the change of variables and exp-product transformations such that $L_{B} {f \to}_{C} {r \to}_{E} L_{i n p}$ .

Algorithm 3.1.

General Outline of find_2f1.

INPUT: $L_{i n p} \in Q (x) [\partial]$ and (optional) $a_{f} m a x$ where
1. $L_{i n p} =$ a second order regular singular irreducible operator,
2. $a_{f} m a x =$ bound for the algebraic degree $a_{f}$ (See Example 1.2). If omitted, then $a_{f} m a x = 2$ which means our implementation tries $a_{f} = 1$ and $a_{f} = 2$ .
OUTPUT: Solutions of $L_{i n p}$ in the form of (1), or an empty list.
For each $a_{f} \in {1, \dots, a_{f} m a x}$ :
Use Section 3.4 to compute candidates for $L_{B}$ and $d_{f}$ . This is the combinatorial part of the algorithm.
For a candidate $(L_{B}, d_{f})$ , compute formal solutions of $L_{B}$ and $L_{i n p}$ at a non-removable singularity (see Remark 2.2 in Section 2.3) up to precision $a \geq 2 (a_{f} + 1) (d_{f} + 1) + 6$ . Take the quotients of formal solutions and compute series expansions for $q^{- 1}$ and $Q$ which will be used to compute

$f = q^{- 1} (c Q (x))$ (7)

in the next step.
Choose a good prime number $ℓ$ and try to find $c$ mod $ℓ$ by looping $c = 1, 2, \dots, ℓ - 1$ as in Section 3.5. For each $c$ :
1. Compute $f$ mod $(x^{a}, ℓ)$ from equation (7) and use it to reconstruct $f$ mod $ℓ$ (the image of $f$ in $F_{ℓ} (x)$ ). If it fails for every $c$ , then proceed with the next candidate GHDO (if any) in Step 2. If no candidates remain, then return an empty list.
2. If rational reconstruction in Step 3.1 succeeds for some $c$ values, then apply Hensel lifting (Section 3.6) to find $f$ mod a power of $ℓ$ . Then try rational number reconstruction. If it does not fail for at least one $c$ value, then we have $f$ . If no solution is found (see Remark 3.4 in Section 3.6.1), then proceed with the next candidate GHDO (if any) in Step 2. If no candidates remain, then return an empty list.
3. Use Section 2.3 to compute the parameter $r$ of the exp-product transformation.
Return a basis of $_{2} F_{1}$ -type solutions of $L_{i n p}$ .

Step 2 is explained in Sections 3.2 and 3.4. Step 3 is the quotient method, see Section 3.5 for more. Steps 3.2 and 3.3 are explained in Sections 3.6 and 3.7 respectively. A Maple implementation of Algorithm 3.1 and some examples can be found at Imamoglu (2015a).

3.2 Degree Bounds for Pullback Functions

Theorem 3.1 (Riemann-Hurwitz Formula).

Let $X$ and $Y$ be two algebraic curves with genera $g_{X}$ and $g_{Y}$ respectively. If $f : X ⟶ Y$ is a non-constant morphism, then

2 g_{X} - 2 = deg (f) (2 g_{Y} - 2) + \sum p \in X (e_{p} - 1) .

(8)

Here $e_{p}$ denotes the ramification order at $p \in X$ . See Hartshorne (1977) for more details.

Let $L_{B} \in Q (x) [\partial]$ be a GHDO, $d_{f} := deg (f)$ , and assume that

L_{B} {f : P^{1} ⟼ P^{1} - ----- \to}_{C} {r \to}_{E} L_{i n p} .

Section 3.1 of Imamoglu and van Hoeij (2015) gives an a priori bound for $d_{f}$ ,

d_{f} \leq {\begin{matrix} 6 (n_{t r u e} - 2), & logarithmic case, 36 (n_{t r u e} - \frac{7}{3}), & non-logarithmic case. \end{matrix}

(9)

where $n_{t r u e}$ is the number of true singularities of $L_{i n p}$ . Algorithm 3.1 uses this only as an initial degree bound.

3.3 Riemann-Hurwitz Type Formula For Differential Equations

Remark 3.1.

Let $X$ be any algebraic curve and $C (X)$ be its function field. The ring $D_{C (X)} := C (X) [\partial_{t}]$ is the ring of differential operators on $X$ . Here $t \in C (X) ∖ C$ . An element $L \in D_{C (X)}$ is a differential operator defined on the algebraic curve $X$ .

Theorem 3.2.

(Baldassari and Dwork, 1979, Lemma 1.5) Let $X$ , $Y$ be two algebraic curves with genera $g_{X}$ , $g_{Y}$ , and function fields $C (X)$ , $C (Y)$ respectively. Let $f : X ⟶ Y$ be a non-constant morphism. The morphism $f$ corresponds to a homomorphism $C (Y) ⟶ C (X)$ , which in turn corresponds to a homomorphism $D_{C (Y)} ⟶ D_{C (X)}$ . If $L_{1} \in D_{C (Y)}$ with $o r d (L_{1}) = 2$ and $L_{2}$ is the corresponding element in $D_{C (X)}$ , then

C o v o l (L_{2}, X) = deg (f) \cdot C o v o l (L_{1}, Y)

(10)

where

C o v o l (L, X) := 2 g_{X} - 2 + \sum p \in X (1 - Δ (L, p))

and where $Δ (L, p)$ is the absolute value of the exponent difference of $L$ at $p$ .

Proof..

Following Baldassari and Dwork (1979), take finite sets $S \subseteq Y$ and $T = f^{- 1} (S)$ in such a way that all singularities of $L_{1}$ are in $S$ , all singularities of $L_{2}$ are in $T$ , and all branching points in $X$ are in $T$ as well.

$# T = \sum p \in T 1$	$= \sum p \in T e_{p} + \sum p \in T (1 - e_{p})$	(11)
	$= deg (f) \cdot # S + \sum p \in X (1 - e_{p})$	(12)
	$= deg (f) \cdot # S - (2 g_{X} - 2 - deg (f) (2 g_{Y} - 2)) .$	(13)

From (12) to (13) we used (8). Then,

$\sum p \in X (1 - Δ (L_{2}, p))$	$= \sum p \in T (1 - Δ (L_{2}, p))$	(14)
	$= \sum p \in T 1 - \sum p \in T Δ (L_{2}, p)$	(15)
	$= # T - deg (f) \sum s \in S Δ (L_{1}, s) .$	(16)

Then, combine (13) and (16) and get

2 g_{X} - 2 + \sum p \in X (1 - Δ (L_{2}, p)) = deg (f) (2 g_{Y} - 2 + \sum s \in Y (1 - Δ (L_{1}, s)))

(17)

which is the same as (10). ∎

Corollary 3.1.

Let $X = Y = P^{1}$ and suppose that $L_{B} {f : P^{1} \to P^{1} - ---- \to}_{C} {r \to}_{E} L_{i n p}$ where $L_{B} \in C (x) [\partial]$ is a GHDO with exponent differences $[α_{0}, α_{1}, α_{\infty}]$ at ${0, 1, \infty}$ . Since an exp-product transformation does not affect exponent differences, Theorem 3.2 gives the following equation for $C o v o l (L_{i n p}, P^{1})$ :

- 2 + \sum p \in P^{1} (1 - Δ (L_{i n p}, p)) = deg (f) ⎛ ⎝ - 2 + \sum i \in {0, 1, \infty} (1 - α_{i}) ⎞ ⎠ .

(18)

Corollary 3.2.

Let $L_{B}$ and $L_{i n p}$ be as in Corollary 3.1. Both have rational function coefficients. This time, suppose that $f, r$ in $L_{B} {f \to}_{C} {r \to}_{E} L_{i n p}$ are algebraic functions. Then $f : X \to P^{1}$ for an algebraic curve $X$ whose function field $C (X) = C (x, f)$ is an algebraic extension of both $C (x) ≅ C (P^{1})$ and $C (f) ≅ C (P^{1})$ . Let $a_{f}$ and $d_{f}$ denote the degrees of these extensions.

Applying (10) to both field extensions gives:

C o v o l (L_{i n p}, P^{1}) = \frac{d_{f}}{a_{f}} ⎛ ⎝ - 2 + \sum i \in {0, 1, \infty} (1 - α_{i}) ⎞ ⎠ .

(19)

3.4 Candidate Exponent Differences

This section explains how to obtain exponent differences for candidate GHDOs.

Algorithm 3.2.

General Outline of find_expdiffs.

INPUT: $e_{i n p}$ , $e_{a p p}$ , and $a_{f}$ where
1. [label=()]
2. $e_{i n p} =$ the list of exponent differences of $L_{i n p}$ at its true singularities,
3. $e_{r e m} =$ the (possibly empty) list of exponent differences of $L_{i n p}$ at its removable singularities,
4. $a_{f}$ = candidate algebraic degree.
OUTPUT: A list of all lists $e_{B} = [α_{0}, α_{1}, α_{\infty}, d]$ of integers or rational numbers where $[α_{0}, α_{1}, α_{\infty}]$ is a list of candidate exponent differences and $d$ is a candidate degree $d_{f}$ for $f$ such that:
1. [label=()]
2. For every exponent difference $m$ in $e_{i n p}$ there exists $e \in Q$ with $e \cdot a_{f} \in {1, \dots, d}$ such that $m = e \cdot α_{i}$ for some $i \in {0, 1, \infty}$ .
3. The multiplicities $e$ are consistent with (8), and their sums are compatible with $d$ , see the last paragraph in Step $2$ .
Let ${¯ ¯¯ ¯ α}_{1}, {¯ ¯¯ ¯ α}_{2}, {¯ ¯¯ ¯ α}_{3}$ $=$ $α_{0}, α_{1}, α_{\infty}$ . After reordering we may assume that ${¯ ¯¯ ¯ α}_{1}$ , $\dots$ , ${¯ ¯¯ ¯ α}_{k}$ $\in$ $Z$ and ${¯ ¯¯ ¯ α}_{k + 1}, \dots, {¯ ¯¯ ¯ α}_{3} \notin Z$ for $k \in {0, 1, 2, 3}$ . For each $k \in {0, 1, 2, 3}$ we use CoverLogs in Imamoglu (2015a) to compute candidates for ${¯ ¯¯ ¯ α}_{1}, \dots, {¯ ¯¯ ¯ α}_{k} \in Z$ .
Algorithm CoverLogs computes candidates that meet these requirements:
- Logarithmic singularities are true singularities with integer exponent differences. If $L_{i n p}$ has at least one logarithmic singularity $s$ with exponent difference $Δ (L_{i n p}, s)$ , then a candidate $L_{B}$ must have at least one logarithmic singularity; at least one of the ${¯ ¯¯ ¯ α}_{1}, {¯ ¯¯ ¯ α}_{2}, {¯ ¯¯ ¯ α}_{3}$ must be an integer that divides $a_{f} \cdot Δ (L_{i n p}, s)$ , and for every ${¯ ¯¯ ¯ α}_{i} \in Z$ there must be at least one $s$ such that ${¯ ¯¯ ¯ α}_{i}$ divides $a_{f} \cdot Δ (L_{i n p}, s)$ .
- $Δ (L_{i n p}, s) = 0$ for some $s$ $⟺$ $0 \in {{¯ ¯¯ ¯ α}_{1}, {¯ ¯¯ ¯ α}_{2}, {¯ ¯¯ ¯ α}_{3}}$ .
- Theorem 2.1.
If ${¯ ¯¯ ¯ α}_{1} + \dots + {¯ ¯¯ ¯ α}_{k} \neq 0$ , then algorithm CoverLogs also computes the exact degree $d_{f}$ of $f$ using Theorem 2.1 which shows that $d_{f} ({¯ ¯¯ ¯ α}_{1} + \dots + {¯ ¯¯ ¯ α}_{k}) / a_{f}$ must be the sum of the logarithmic exponent differences of $L_{i n p}$ . Otherwise, it uses (9) to compute a bound for $d_{f}$ , and uses it as $d_{f}$ to compute a candidate degree.
We will explain only the case $a_{f} = 1$ , and only $k = 1$ , which is the case $[{¯ ¯¯ ¯ α}_{1}, {¯ ¯¯ ¯ α}_{2}, {¯ ¯¯ ¯ α}_{3}] = [α_{0}, α_{1}, α_{\infty}]$ , where $α_{0} \in Z$ and $α_{1}, α_{\infty} \notin Z$ .
Let $k = 1$ . Let $α_{0} \in Z$ be one of the candidates from algorithm CoverLogs. We need to find candidates for $α_{1}$ and $α_{\infty}$ .
The logarithmic singularities of $L_{i n p}$ come from the point $0$ . Non-integer exponent differences of $L_{i n p}$ must be multiples of $α_{1}$ or $α_{\infty}$ . Let $S_{N}$ be the set of non-logarithmic exponent differences of $L_{i n p}$ and $S_{R}$ be the set of exponent differences of $L_{i n p}$ at its removable singularities. Consider the set

$Γ_{1} = {\begin{matrix} Γ_{A} = {\frac{max (S_{N})}{b} : b = 1, \dots, d_{f}} & if S_{N} \neq \emptyset, Γ_{B} = {\frac{a}{b} : a \in S_{R} \cup {1}, b = 1, \dots, d_{f}} & otherwise% . \end{matrix}$

$α_{1}$ (or $α_{\infty}$ , but if so, we may interchange them) must be one of the elements of $Γ_{1}$ . We loop over all elements of $Γ_{1}$ . Assume that a candidate for $α_{1}$ is chosen. Let $Ω = S_{N} ∖ α_{1} Z$ . Now consider the set

$Γ_{\infty} = {\begin{matrix} Γ_{A} \cup Γ_{B} & if Ω = \emptyset, {\frac{g}{b} : g = gcd (Ω) : b = 1, \dots, d_{f}} & otherwise . \end{matrix}$
Now take all pairs $(α_{\infty}, d)$ satisfying (19), $α_{\infty} \in Γ_{\infty}$ , $1 \leq d \leq d_{f}$ , with additional restrictions on $d$ , as follows:
For every potential non-zero value $v$ for one of the $α_{i}$ ’s we pre-compute a list of integers $N_{v}$ by dividing all exponent differences of $L_{i n p}$ by $v$ and then selecting the quotients that are integers. Next, let $D_{v}$ be the set of all $1 \leq d \leq d_{f}$ that can be written as the sum of a sublist of $N_{v}$ . Each time a non-zero value $v$ is taken for one of the $α_{i}$ , it imposes the restriction $d \in D_{v}$ . This means that we need not run a loop for $α_{\infty} \in Γ_{\infty}$ , instead, we run a (generally much shorter) loop for $d$ (taking values in the intersection of the $D_{v}$ ’s so far) and then for each such $d$ compute $α_{\infty}$ from (19). We also check if $d \in D_{α_{\infty}}$ .
Return the list of candidate exponent differences with a candidate degree, the list of lists $[α_{0}, α_{1}, α_{\infty}, d]$ , for candidate GHDOs.

3.5 Quotient Method

In this section, we explain a method to recover the pullback function $f$ . We will explain our algorithm for rational pullback functions. For algebraic pullback functions, the only difference is the lifting algorithm, which is explained in Section 3.6. Note that we can always compute the formal solutions of a given differential equation $L_{i n p} (y) = 0$ up to a finite precision.

3.5.1 Non-logarithmic Case

Let the second order differential equation $L_{i n p} (y) = 0$ be given. Let $L_{B}$ be a GHDO such that $L_{B} {f \to}_{C} {r \to}_{E} L_{i n p} .$ Let $f : P_{x}^{1} \mapsto P_{z}^{1}$ and $L_{1} {f \to}_{C} L_{2}$ . If $x = p$ is a singularity of $L_{2}$ and $z = s$ is a singularity of $L_{1}$ , then we say that $p$ comes from $s$ when $f (p) = s$ .

After a change of variables we can assume that $x = 0$ is a singularity of $L_{i n p}$ that comes from the singularity $z = 0$ of $L_{B}$ . This means $f (0) = 0$ and we can write $f = c_{0} x^{v_{0} (f)} (1 + \dots)$ where $c_{0} \in C$ , $v_{0} (f)$ is the multiplicity of $0$ , and the dots refer to an element in $x C [[x]]$ .

Let $y_{1}$ and $y_{2}$ be the formal solutions of $L_{B}$ at $x = 0$ . The following diagram shows the effects of the change of variables and exp-product transformations on the formal solutions of $L_{B}$ ,

y_{i} (x)

{f \to}_{C} y_{i} (f) {r \to}_{E} Y_{i} (x) = exp (\int r d x) y_{i} (f), i \in {1, 2}

where $Y_{1}$ and $Y_{2}$ are solutions of $L_{i n p}$ .

Let $q = \frac{y_{1}}{y_{2}}$ be a quotient of formal solutions of $L_{B}$ . The change of variables transformation sends $x$ to

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1 Introduction

Example 1.1 (Rational Pullback Function).

Remark 1.1 (The Quotient Method).

Remark 1.2.

Example 1.2 (Algebraic Pullback Function).

Example 1.3 (Finding Solutions in the form of (2) using an Integral Basis).

2 Preliminaries

2.1 Differential Operators, Singularities, Formal Solutions

Definition 2.1.

2.2 Gauss Hypergeometric Differential Operator and $_{2} f_{1}$ Function

2.3 Transformations and Singularities

Remark 2.1.

Remark 2.2.

Remark 2.3.

Theorem 2.1.

3 Computing Solutions of a Second Order Linear Differential Operator in the form of (1) by using Quotients of Formal Solutions

3.1 Problem Statement

Algorithm 3.1.

3.2 Degree Bounds for Pullback Functions

Theorem 3.1 (Riemann-Hurwitz Formula).

3.3 Riemann-Hurwitz Type Formula For Differential Equations

Remark 3.1.

Theorem 3.2.

Proof..

Corollary 3.1.

Corollary 3.2.

3.4 Candidate Exponent Differences

Algorithm 3.2.

3.5 Quotient Method

3.5.1 Non-logarithmic Case

Computing Hypergeometric Solutions of Second Order Linear Differential Equations using Quotients of Formal Solutions and Integral Bases

Authors

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This week in AI

1 Introduction

Example 1.1 (Rational Pullback Function).

Remark 1.1 (The Quotient Method).

Remark 1.2.

Example 1.2 (Algebraic Pullback Function).

Example 1.3 (Finding Solutions in the form of (2) using an Integral Basis).

2 Preliminaries

2.1 Differential Operators, Singularities, Formal Solutions

Definition 2.1.

2.2 Gauss Hypergeometric Differential Operator and 2f1 Function

2.3 Transformations and Singularities

Remark 2.1.

Remark 2.2.

Remark 2.3.

Theorem 2.1.

3 Computing Solutions of a Second Order Linear Differential Operator in the form of (1) by using Quotients of Formal Solutions

3.1 Problem Statement

Algorithm 3.1.

3.2 Degree Bounds for Pullback Functions

Theorem 3.1 (Riemann-Hurwitz Formula).

3.3 Riemann-Hurwitz Type Formula For Differential Equations

Remark 3.1.

Theorem 3.2.

Proof..

Corollary 3.1.

Corollary 3.2.

3.4 Candidate Exponent Differences

Algorithm 3.2.

3.5 Quotient Method

3.5.1 Non-logarithmic Case

2.2 Gauss Hypergeometric Differential Operator and $_{2} f_{1}$ Function