IX.
HYPERGEOMETRIC EQUATIONS and THEIR APPLICATIONS
9.3 DIFFERENTIAL EUQATION
The hypergeometric differential equation is a second-order linear differential equation whose
solutions are hypergeometric functions. It's fundamental in mathematical physics and appears in
problems with special symmetries.
General Form:
2
d y dy
x (1−x ) 2
+ [ c− ( a+b+1 ) x ] −aby =0
d x dx
where a,b,c are constants.
The standard solution is the hypergeometric function:
y(x) = 2F1 (a;b;c;x)
Example.
1. Prove that ( 1−x )−∞=F (α , β , β , x) where we have hypergeometric function
Practice Problems
1. Identify the values of a,b,ca, b, ca,b,c for the equation:
x (1−x ) y ' '+[3−5 x ] y '−6 y=0
2. Find the general solution of:
x (1−x ) y ' '+(4−6 x ) y '−8 y=0
3. Prove that
IX. HYPERGEOMETRIC EQUATIONS and THEIR APPLICATIONS
xF(1,1,2;-x) = log(1+x)
9.4 GKZ SYSTEMS
The GKZ hypergeometric system (or A-hypergeometric system) is a system of partial
differential equations introduced by Gel'fand, Kapranov, and Zelevinsky. It generalizes the
classical hypergeometric differential equations to to multiple variables and higher dimensions.
These systems arise from toric geometry and are connected to algebraic geometry, representation
theory, and mathematical physics. A GKZ hypergeometric system is denoted by:
HA ( β)
Where:
d xn
A ∈ Z :integer ¿ full rank
d
β ∈ C :complex parameter vector
IX. HYPERGEOMETRIC EQUATIONS and THEIR APPLICATIONS
9.5 REGULAR SINGULARITIES
In the context of ordinary differential equations (ODEs), a singular point x0x_0x0 of an
equation is a point where the coefficients of the equation become singular (i.e., not analytic). If
the solution of the equation can still be expressed in a power series expansion near this point
(with some conditions), the singularity is said to be a regular singularity.
Definition: Regular Singular Point
A point x0x_0x0 is a regular singular point of the differential equation:
IX. HYPERGEOMETRIC EQUATIONS and THEIR APPLICATIONS
9.6 THE FROBENIUS METHOD
IX. HYPERGEOMETRIC EQUATIONS and THEIR APPLICATIONS
9.7 THE MULTIVARIATE LOGARITHMIC SERIES
The Multivariate Logarithmic Series Distribution (MLSD) is a generalization of the logarithmic series
distribution to multiple variables. It is often used in ecology, linguistics, and contingency tables. Its
probability mass function (PMF) involves generalized hypergeometric series.
IX. HYPERGEOMETRIC EQUATIONS and THEIR APPLICATIONS
9.8THE INDICIAL IDEAL
In the study of AAA-hypergeometric systems (also called GKZ systems, after Gelfand, Kapranov, and
Zelevinsky), solutions are often studied using the tools of algebraic D-modules. The indicial ideal plays a
central role in determining the behavior of solutions near singularities (e.g., around coordinate
hyperplanes).
IX. HYPERGEOMETRIC EQUATIONS and THEIR APPLICATIONS
9.9FAKE INDICIAL IDEAL
The fake indicial ideal is an approximation of the true indicial ideal. It is formed by ignoring the initial
terms of the toric operators and only focusing on the Euler operators and the leading monomials of the
toric ideal.
IX. HYPERGEOMETRIC EQUATIONS and THEIR APPLICATIONS
9.10 STANDARD PAIRS
Standard pairs come from the theory of monomial ideals and are used in describing solutions to AAA-
hypergeometric systems.
IX. HYPERGEOMETRIC EQUATIONS and THEIR APPLICATIONS
9.11 LOGARITHMIC FREE HYPERGEOMETRIC SERIES
The fake indicial ideal is an approximation of the true indicial ideal. It is formed by ignoring the initial
terms of the toric operators and only focusing on the Euler operators and the leading monomials of the
toric ideal.
IX. HYPERGEOMETRIC EQUATIONS and THEIR APPLICATIONS
9.12RATIONAL HYPERGEOMETRIC FUNCTIONS
A rational hypergeometric function is a solution to an AAA-hypergeometric system that is a rational
function in the variables
