HypergeometricUStar

The regularized Tricomi confluent hypergeometric function $U^*(a,b,z) = z^a\,U(a,b,z)$, for $b$ not an integer. Provided by @enumeratio/analytic.

✓ checked against that system · ! checked, with disagreements (hover a chip)FungrimHypergeometricUStar✓↗Fungrim entry

HypergeometricUStar(a, b, z)$z^a$ times Tricomi's confluent hypergeometric $U(a,b,z)$.

Typed signatures

Type	Package	Applies
(number, number, number) -> number	analytic	always

Domain: Special functions

Details

• compute-engine declares HypergeometricU but does not evaluate it numerically (a symbolic stub only), so $U$ itself is supplied here via Kummer's connection formula in terms of the entire confluent hypergeometric $M = {}_1F_1$: $U(a,b,z) = \tfrac{\Gamma(1-b)}{\Gamma(a-b+1)}M(a,b,z) + \tfrac{\Gamma(b-1)}{\Gamma(a)}z^{1-b}M(a-b+1,2-b,z)$.
• $1/\Gamma$ is taken directly (zero at the nonpositive integers) rather than as a raw division, so the formula stays finite exactly where $a-b+1$ or $a$ lands on one of $\Gamma$'s poles — not an edge case Fungrim's own identities avoid.
• $b$ at (or very near) an integer is declined: both $\Gamma(1-b)$ and $\Gamma(b-1)$ blow up there, and the log-case limit that resolves it is not implemented. So is any point where the two terms of the formula cancel past a relative error of $10^{-11}$ (a $b$ within about $10^{-3}$ of an integer, or a large $z$): measured against mpmath on a grid of 546 real points, every answered point is within $5\times10^{-12}$.
• $z^{1-b}$ and the closing $z^a$ take the principal branch, matching mpmath's hyperu and Wolfram's HypergeometricU.

Examples

4 more kept as data, checked against the oracles

Basic #

$U^*(\tfrac12,\tfrac32,2) = 1$

live ↗

In$\mathrm{chop}(N(\mathrm{HypergeometricUStar}(1/2, 3/2, 2))-1)$Out$0.0$

a generic point, checked against mpmath's hyperu

live ↗

In$\mathrm{HypergeometricUStar}(1.3, 2.7, 4)$Out$1.113\,705\,290\,586\,790\,6$

Possible issues #

integer $b$ stays symbolic — the connection formula's log-case limit is not implemented

live ↗

In$\mathrm{HypergeometricUStar}(1, 2, 3)$Out$\mathrm{HypergeometricUStar}(1, 2, 3)$

About

Examples	7
Crosswalk	FungrimFungrim entry ×11

Implementation

primitive · numericirreducibly numeric — it evaluates rather than rewrites

nativeenginetypescriptpackages/symbols/analysis/analytic/src/hypergeometric-ustar.ts

mappedexternalwolfram / mpmathpackages/oracle/src/mappings.ts

z^a HypergeometricU[a,b,z]; z**a * mpmath.hyperu(a,b,z).