Hypergeometric function

Key takeaways

Hypergeometric function belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Hypergeometric function to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Hypergeometric function from memory before moving on to harder problems.

Reference excerpt

In mathematics, the Gaussian or ordinary hypergeometric function 2F1(a, b; c; z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation. For systematic lists of some of the many thousands of published identities involving the hypergeometric function, see the reference works by Erdélyi et al. (1953) and Olde Daalhuis (2010). There is no known system for organizing all of the identities; indeed, there is no known algorithm that can generate all identities; a number of different algorithms are known that generate different series of identities. The theory of the algorithmic discovery of identities remains an active research topic.

History The term "hypergeometric series" was first used by John Wallis in his 1655 book Arithmetica Infinitorum. Hypergeometric series were studied by Leonhard Euler, but the first full systematic treatment was given by Carl Friedrich Gauss (1813). Studies in the nineteenth century included those of Ernst Kummer (1836), and the fundamental characterisation by Bernhard Riemann (1857) of the hypergeometric function by means of the differential equation it satisfies. Riemann showed that the second-order differential equation for 2F1(z), examined in the complex plane, could be characterised (on the Riemann sphere) by its three regular singularities. The cases where the solutions are algebraic functions were found by Hermann Schwarz (Schwarz's list).

The hypergeometric series The hypergeometric function is defined for |z| < 1 by the power series

2 F 1 ( a , b ; c ; z ) = ∑ n = 0 ∞ ( a ) n ( b ) n ( c ) n z n n ! = 1 + a b c z 1 ! + a ( a + 1 ) b ( b + 1 ) c ( c + 1 ) z 2 2 ! + ⋯ . {\displaystyle {}_{2}F_{1}(a,b;c;z)=\sum _{n=0}^{\infty }{\frac {(a)_{n}(b)_{n}}{(c)_{n}}}{\frac {z^{n}}{n!}}=1+{\frac {ab}{c}}{\frac {z}{1!}}+{\frac {a(a+1)b(b+1)}{c(c+1)}}{\frac {z^{2}}{2!}}+\cdots .}

It is undefined (or infinite) if c equals a non-positive integer. Here (q)n is the (rising) Pochhammer symbol, which is defined by:

( q ) n = { 1 n = 0 q ( q + 1 ) ⋯ ( q + n − 1 ) n > 0 {\displaystyle (q)_{n}={\begin{cases}1&n=0\\q(q+1)\cdots (q+n-1)&n>0\end{cases}}}

The series terminates if either a or b is a nonpositive integer, in which case the function reduces to a polynomial:

2 F 1 ( − m , b ; c ; z ) = ∑ n = 0 m ( − 1 ) n ( m n ) ( b ) n ( c ) n z n . {\displaystyle {}_{2}F_{1}(-m,b;c;z)=\sum _{n=0}^{m}(-1)^{n}{\binom {m}{n}}{\frac {(b)_{n}}{(c)_{n}}}z^{n}.}

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Hypergeometric function

Start with the simplest possible case. Write down what Hypergeometric function claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Hypergeometric function before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Hypergeometric function ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Hypergeometric function

In research: Hypergeometric function appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Hypergeometric function in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Hypergeometric function is common in secondary-school and first-year university syllabi. It links to neighbouring topics Factorial and binomial topics, Hypergeometric functions, Ordinary differential equations, so understanding it makes those chapters shorter.
In everyday life: Look for Hypergeometric function outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Hypergeometric function in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hypergeometric function means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Hypergeometric function out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hypergeometric function in simple terms?

In mathematics, the Gaussian or ordinary hypergeometric function 2F1(a, b; c; z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE).

Why does Hypergeometric function matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Hypergeometric function?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Hypergeometric function.