Gauss hypergeometric equation

The Gauss equation has singularities at $0,1,\infty$ and normalized exponent-zero solution $F(A,B;C;u)$ at the origin.

For the Gauss hypergeometric equation with parameters $\alpha ,\beta ,\gamma$, a Frobenius solution $y=x^{\sigma }{\sum }_{n\ge 0}{a}_n{x}^n$, $a_0\ne 0$, has indicial roots $\sigma =0$ and $\sigma =1-\gamma$. Matching each successive power gives the displayed undivided recurrence. The usual ratio formula is valid only when its left coefficient is nonzero. At integer-separated roots, a zero coefficient requires a compatibility check as in an undetermined coefficient at Frobenius resonance; otherwise a logarithmic companion may be necessary. For example, $\gamma =0,\sigma =0$ gives $0=\alpha \beta a_0$ at $n=1$, obstructing that exponent-zero series when $\alpha \beta \ne 0$.

Multiply a solution with parameters $(c-a,c-b;c)$ by $(1-z{)}^{c-a-b}$. Its local exponents become exactly those of the Gauss hypergeometric equation with parameters $(a,b;c)$: those at one shift by $c-a-b$, and those at infinity shift by its negative. Direct differentiation gives the same differential equation. Near zero, choose the factor with value one; uniqueness of the analytic normalized solution, for $c$ not a nonpositive integer, gives the identity. Compatible analytic continuation extends it to other domains and parameters where the normalized functions are defined.

Put $d=c-a-b\notin \mathbb{Z}$ and choose a branch of $(1-z{)}^d$. The solution regular at zero isThe two characteristic exponents at a regular singular point are $0,d$. The Euler integral for the hypergeometric function determines the first coefficient by a finite endpoint limit and the second by scaling the singular endpoint; analytic continuation extends the parameter ranges.

The Gauss hypergeometric function is the normalized analytic solution at zero,initially for $|z|<1$ and $c\notin \{0,-1,-2,\dots \}$, followed by analytic continuation where possible.

The ordinary hypergeometric function is ${}_2{F}_1(a,b;c;z)={\sum }_{n\ge 0}(a{)}_n(b{)}_n{z}^n/((c{)}_{n}n!)$ near zero for admissible $c$. It solves the Gauss hypergeometric equation and extends by analytic continuation.

For $\Re c>\Re b>0$ and a fixed branch away from the cut,Expand $(1-zs{)}^{-a}$ for $|z|<1$ and integrate termwise using the beta function to recover the Gauss hypergeometric function series. Analytic continuation extends the identity in $z$.

With compatible branches and initially near $z=0$,It follows by applying a Möbius transformation and a dependent-variable rescaling to the same Papperitz symbol, then matching the normalized analytic solution at zero.

When $C$ is not an integer, a second local solution at zero is $u^{1-C}F(A-C+1,B-C+1;2-C;u)$.

Away from resonant parameter cases, two independent local solutions at infinity areand the expression obtained by exchanging $a$ and $b$. Every analytic continuation of a hypergeometric solution into their common domain is a constant linear combination of this basis.

The binomial series gives $F(A,C;C;u)=(1-u{)}^{-A}$ wherever the defining series converges, followed by analytic continuation.

Special parameter choices can identify a hypergeometric solution with an elementary solution of the same second-order differential equation by matching its local normalization.

For compatible local branches near $x=0$,It follows by transforming the harmonic oscillator equation $y^{\prime \prime }+k^{2}y=0$ into the Gauss hypergeometric equation and selecting the solution with value one and derivative zero at the origin.

For $k\ne 0$ and compatible local branches near $x=0$,This is obtained from the second local hypergeometric solution at zero and its normalization.