Gauss hypergeometric function
E596519
The Gauss hypergeometric function is a special function defined by a power series that generalizes many elementary and higher transcendental functions and plays a central role in mathematical analysis, differential equations, and mathematical physics.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Gauss hypergeometric function canonical | 2 |
| Gaussian hypergeometric function | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6482510 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Gauss hypergeometric function Context triple: [Pochhammer symbol, usedIn, Gauss hypergeometric function]
-
A.
Gamma function
The Gamma function is a fundamental extension of the factorial function to complex and real non-integer arguments, widely used in analysis, probability, and mathematical physics.
-
B.
Mittag-Leffler function
The Mittag-Leffler function is a complex function that generalizes the exponential function and plays a central role in fractional calculus and the theory of differential and integral equations.
-
C.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
-
D.
Bessel functions
Bessel functions are special mathematical functions that commonly arise as solutions to differential equations with cylindrical symmetry, widely used in physics and engineering.
-
E.
Jacobi elliptic functions
Jacobi elliptic functions are a family of doubly periodic complex functions that generalize trigonometric functions and play a central role in the theory of elliptic integrals and many areas of mathematical physics.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Gauss hypergeometric function Target entity description: The Gauss hypergeometric function is a special function defined by a power series that generalizes many elementary and higher transcendental functions and plays a central role in mathematical analysis, differential equations, and mathematical physics.
-
A.
Gamma function
The Gamma function is a fundamental extension of the factorial function to complex and real non-integer arguments, widely used in analysis, probability, and mathematical physics.
-
B.
Mittag-Leffler function
The Mittag-Leffler function is a complex function that generalizes the exponential function and plays a central role in fractional calculus and the theory of differential and integral equations.
-
C.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
-
D.
Bessel functions
Bessel functions are special mathematical functions that commonly arise as solutions to differential equations with cylindrical symmetry, widely used in physics and engineering.
-
E.
Jacobi elliptic functions
Jacobi elliptic functions are a family of doubly periodic complex functions that generalize trigonometric functions and play a central role in the theory of elliptic integrals and many areas of mathematical physics.
- F. None of above. chosen
Statements (52)
| Predicate | Object |
|---|---|
| instanceOf |
analytic function
ⓘ
hypergeometric function ⓘ special function ⓘ |
| alsoKnownAs |
Gaussian hypergeometric function
ⓘ
linked to:
Gauss hypergeometric function
ordinary hypergeometric function ⓘ |
| appearsIn |
conformal field theory
ⓘ
general relativity ⓘ potential theory ⓘ probability theory ⓘ quantum mechanics ⓘ solutions of second-order linear ordinary differential equations with three regular singular points ⓘ statistics ⓘ |
| belongsTo | hypergeometric series family {}_pF_q ⓘ |
| convergesFor | |z|<1 for general complex parameters a,b,c with c not a nonpositive integer ⓘ |
| definedBySeries | {}_2F_1(a,b;c;z)=\sum_{n=0}^{\infty}\frac{(a)_n(b)_n}{(c)_n}\frac{z^n}{n!} ⓘ |
| dependsOnParameters |
a
ⓘ
b ⓘ c ⓘ |
| dependsOnVariable | z ⓘ |
| domain | complex variable z with analytic continuation to \mathbb{C}\setminus[1,\infty) ⓘ |
| generalizes |
Bessel functions
ⓘ
Chebyshev polynomials ⓘ
linked to:
Chebyshev polynomials of the first kind
Gegenbauer polynomials ⓘ Jacobi polynomials ⓘ Legendre functions ⓘ
linked to:
Legendre polynomials
arcsin function ⓘ arctan function ⓘ binomial series (1-z)^{-a} ⓘ logarithm function ⓘ |
| hasBranchPoints |
z=1
ⓘ
z=\infty ⓘ |
| hasConnectionFormula | linear relations between values at z,1-z,1/z ⓘ |
| hasIntegralRepresentation | {}_2F_1(a,b;c;z)=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int_0^1 t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a}\,dt for suitable parameters ⓘ |
| hasSymmetry | {}_2F_1(a,b;c;z)={}_2F_1(b,a;c;z) ⓘ |
| hasTransformationFormula |
Euler transformation {}_2F_1(a,b;c;z)=(1-z)^{c-a-b}{}_2F_1(c-a,c-b;c;z)
ⓘ
Pfaff transformation {}_2F_1(a,b;c;z)=(1-z)^{-a}{}_2F_1\left(a,c-b;c;\frac{z}{z-1}\right) ⓘ |
| isCaseOf | generalized hypergeometric function with p=2,q=1 ⓘ |
| namedAfter | Carl Friedrich Gauss ⓘ |
| parameterDomain | complex parameters a,b,c with c not in \{0,-1,-2,\dots\} ⓘ |
| reducesTo | polynomial when a or b is a nonpositive integer ⓘ |
| relatedTo |
beta function
ⓘ
gamma function ⓘ |
| satisfiesDifferentialEquation | z(1-z)y''+[c-(a+b+1)z]y'-aby=0 ⓘ |
| specialValue |
{}_2F_1(a,b;a;z)=(1-z)^{-b}
ⓘ
{}_2F_1(a,b;b;z)=(1-z)^{-a} ⓘ {}_2F_1(a,b;c;0)=1 ⓘ |
| symbol | {}_2F_1(a,b;c;z) ⓘ |
| usedFor |
analytic continuation of many special functions
ⓘ
asymptotic analysis ⓘ evaluation of definite integrals ⓘ representation of orthogonal polynomials ⓘ |
| usesNotation | (q)_n for the Pochhammer symbol ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Gauss hypergeometric function Description of subject: The Gauss hypergeometric function is a special function defined by a power series that generalizes many elementary and higher transcendental functions and plays a central role in mathematical analysis, differential equations, and mathematical physics.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Gauss hypergeometric function