Appendix/Mathematics/Hypergeometric

Hypergeometric Differential Equation

According to §9.151 (p. 1045) of Gradshteyn & Ryzhik (1965), "… a hypergeometric series is one of the solutions of the differential equation,

$0$

$=$

$z(1-z)\frac{d^{2}u}{d{z}^2}+[\gamma -(\alpha +\beta +1)z]\frac{du}{dz}-\alpha \beta u,$

which is called the hypergeometric equation. And, according to §9.10 (p. 1039) of Gradshteyn & Ryzhik (1965), "A hypergeometric series is a series of the form,

$F(\alpha ,\beta ;\gamma ;z)$

$=$

$1+\left[\frac{\alpha \cdot \beta }{\gamma \cdot 1}\right]z+\left[\frac{\alpha (\alpha +1)\beta (\beta +1)}{\gamma (\gamma +1)\cdot 1\cdot 2}\right]z^2+\left[\frac{\alpha (\alpha +1)(\alpha +2)\beta (\beta +1)(\beta +2)}{\gamma (\gamma +1)(\gamma +2)\cdot 1\cdot 2\cdot 3}\right]z^3+\dots$

Among other attributes, Gradshteyn & Ryzhik (1965) note that this, "… series terminates if $\alpha$ or $\beta$ is equal to a negative integer or to zero."

See Also

• 📚 T. E. Sterne (1937, MNRAS, Vol. 97, pp. 582 - 593): Models of Radial Oscillation
• 📚 C. Prasad (1948, MNRAS, Vol. 108, pp. 414 - 416): Radial Oscillations of a Particular Stellar Model
• Z. Kopal (1948, Proceedings of the National Academy of Sciences, Vol. 34, pp. 377 - 384, Radial Oscillations of the Limiting Models of Polytropic Gas Spheres.
• In an article titled, "Radial Oscillations of a Stellar Model," C. Prasad (1949, MNRAS, 109, 103) investigated the properties of an equilibrium configuration with a prescribed density distribution given by the expression,

  $\rho (r)={\rho }_c[1-(\frac{r}{R}{)}^2],$

  where, ${\rho }_c$ is the central density and, $R$ is the radius of the star.

• R. Van der Borght (1970, Proceedings of the Astronomical Society of Australia, Vol. 1, Issue 7, pp. 325 - 326), Adiabatic Oscillations of Stars.