Hypergeometric Function

A Generalized Hypergeometric Function ${}_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;x)$ is a function which can be defined in the form of a Hypergeometric Series, i.e., a series for which the ratio of successive terms can be written

$${c_{k+1}\over c_k}={P(k)\over Q(k)}={(k+a_1)(k+a_2)\cdots(k+a_p)\over(k+b_1)(k+b_2)\cdots(k+b_q)(k+1)}x.$$ (1)

(The factor of $k+1$ in the Denominator is present for historical reasons of notation.) The function ${}_2F_1(a,b;c;x)$ corresponding to $p=2$, $q=1$ is the first hypergeometric function to be studied (and, in general, arises the most frequently in physical problems), and so is frequently known as ``the'' hypergeometric equation. To confuse matters even more, the term ``hypergeometric function'' is less commonly used to mean Closed Form.

The hypergeometric functions are solutions to the Hypergeometric Differential Equation, which has a Regular Singular Point at the Origin. To derive the hypergeometric function based on the Hypergeometric Differential Equation, plug

$$y = \sum_{n=0}^\infty A_nz^n$$ (2)

$$y' = \sum_{n=0}^\infty nA_nz^{n-1}$$ (3)

$$y'' = \sum_{n=0}^\infty n(n-1)A_nz^{n-2}$$ (4)

into

$$z(1-z)y''+[c-(a+b+1)a]y'-aby=0$$ (5)

to obtain

$\sum_{n=0}^\infty n(n-1)A_n z^{n-1}-\sum_{n=0}^\infty n(n-1)A_nz^n$
$+c\sum_{n=0}^\infty nA_nz^{n-1}+(a+b+1)\sum_{n=0}^\infty nA_nz^n-ab\sum_{n=0}^\infty A_nz^n=0\quad$	(6)

$\sum_{n=2}^\infty n(n-1)A_nz^{n-1}-\sum_{n=0}^\infty n(n-1)A_nz^n$
$\mathop{+}c\sum_{n=1}^\infty nA_nz^{n-1}-(a+b+1)\sum_{n=1}^\infty nA_nz^n-ab\sum_{n=0}^\infty A_nz^n=0\quad$	(7)

$\sum_{n=0}^\infty (n+1)nA_{n+1} z^n-\sum_{n=0}^\infty n(n-1)A_nz^n$
$+c\sum_{n=0}^\infty (n+1)A_{n+1}z^n-(a+b+1)\sum_{n=0}^\infty nA_nz^n-ab\sum_{n=0}^\infty A_nz^n=0\quad$	(8)

$$\sum_{n=0}^\infty [n(n+1)A_{n+1}-n(n-1)A_n+c(n+1)A_{n-1}-(a+b+1)nA_n-abA_n]z^n=0$$ (9)

$$\sum_{n=0}^\infty \left\{{(n+1)(n+c)A_{n+1}-[n(n-1+a+b+1)+ab]A_n}\right\}z^n=0$$ (10)

$$\sum_{n=0}^\infty \{(n+1)(n+c)A_{n+1}-[n^2+(a+b)n+ab]A_n\} z^n=0,$$ (11)

so

$$A_{n+1}={(n+a)(n+b)\over (n+1)(n+c)} A_n$$ (12)

and

$$y=A_0\left[{1+{ab\over 1!c} z + {a(a+1)b(b+1)\over 2!c(c+1)} z^2 + \ldots}\right].$$ (13)

This is the regular solution and is denoted

$${}_2F_1(a,b;c;z) = 1 + {ab\over 1!c} z + {a(a+1)b(b+1)\over 2!c (c+1)} z^2 + \ldots$$

$$= \sum_{n=0}^\infty {(a)_n(b )_n\over (c)_n} {z^n\over n!},$$ (14)

where $(a)_n$ are Pochhammer Symbols. The hypergeometric series is convergent for Real $-1<z<1$, and for $z=\pm 1$ if $c>a+b$. The complete solution to the Hypergeometric Differential Equation is

$$y=A{}_2F_1(a,b;c;z)+Bz^{1-c}{}_2F_1(a+1-c,b+1-c;2-c;z).$$ (15)

Derivatives are given by

$${d\,{}_2F_1(a,b;c;z)\over dz} = {ab\over c}{}_2F_1(a+1,b+1;c+1;z)$$ (16)

$${d^2\,{}_2F_1(a,b;c;z)\over dz^2} = {a(a+1)b(b+1)\over c(c+1)} {}_2F_1(a+2, b+2; c+2; z)$$ (17)

(Magnus and Oberhettinger 1949, p. 8). An integral giving the hypergeometric function is

$${}_2F_1(a,b;c;z) = {\Gamma(c)\over \Gamma(b)\Gamma(c-b)}\int_0^1 {t^{b-1}(1-t)^{c-b-1}\over (1-tz)^a}\,dt$$ (18)

as shown by Euler in 1748.

A hypergeometric function can be written using Euler's Hypergeometric Transformations

$$t \to t$$ (19)

$$t \to 1-t$$ (20)

$$t \to (1-z-tz)^{-1}$$ (21)

$$t \to {1-t\over 1-tz}$$ (22)

in any one of four equivalent forms

$${}_2F_1(a,b;c;z) = (1-z)^{-a} \,{}_2F_1(a,c-b;c;z/(z-1))$$ (23)

$$= (1-z)^{-b} \,{}_2F_1(c-a,b;c;z/(z-1))$$ (24)

$$= (1-z)^{c-a-b} \,{}_2F_1(c-a,c-b;c;z).$$ (25)

It can also be written as a linear combination

${}_2F_1(a,b;c;z) ={\Gamma(c)\Gamma(c-a-b)\over\Gamma(c-a)\Gamma(c-b)} {}_2F_1(a,b;a+b+1-c;1-z)$
$\mathop{+}{\Gamma(c)\Gamma(a+b-c)\over \Gamma(a)\Gamma(b)} (1-z)^{c-a-b}_2F_1(c-a,c-b;1+c-a-b;1-z).\quad$	(26)

Kummer found all six solutions (not necessarily regular at the origin) to the Hypergeometric Differential Equation,

$$u_1(x) = {}_2F_1(a,b;c;z)$$

$$u_2(x) = {}_2F_1(a,b;a+b+1-c;1-z)$$

$$u_3(x) = z^{-a}\,{}_2F_1(a,a+1-c;a+1-b;1/z)$$

$$u_4(x) = z^{-b}\,{}_2F_1(b+1-c,b;b+1-a;1/z)$$

$$u_5(x) = z^{1-c}\,{}_2F_1(b+1-c,a+1-c;2-c;z)$$

$$u_6(x) = (1-z)^{c-a-b}\,{}_2F_1(c-a,c-b;c+1-a-b;1-z).$$

Applying Euler's Hypergeometric Transformations to the Kummer solutions then gives all 24 possible forms which are solutions to the Hypergeometric Differential Equation

$$\begin{eqnarray*} u_1^{(1)}(x) &=& {}_2F_1(a,b;c;z)\\ u_1^{(2)}(x) &=& (1-z)^... ...{(4)}(x) &=& z^{c-a-b}(1-z)^{c-a-b}{}_2F_1(1-b,1-a;c+1-a-b;1-z). \end{eqnarray*}$$

Goursat (1881) gives many hypergeometric transformation Formulas, including several cubic transformation Formulas.

Many functions of mathematical physics can be expressed as special cases of the hypergeometric functions. For example,

$${}_2F_1(-l, l+1, 1; (1-z)/2) = P_l(z),$$ (27)

where $P_l(z)$ is a Legendre Polynomial.

$$(1+z)^n={}_2F_1(-n,b;b;-z)$$ (28)

$$\ln(1+z)=z\,{}_2F_1(1,1;2;-z)$$ (29)

Complete Elliptic Integrals and the Riemann P-Series can also be expressed in terms of ${}_2F_1(a,b;c;z)$. Special values include

$\quad {}_2F_1(a,b;a-b+1;-1)= 2^{-a}\sqrt{\pi} {\Gamma(1+a+b)\over \Gamma(1+{\textstyle{1\over 2}}a-b)\Gamma({\textstyle{1\over 2}}+{\textstyle{1\over 2}}a)}$	(30)
$\quad {}_2F_1(1,-a;a;-1) = {\sqrt{\pi}\over 2} {\Gamma(a)\over\Gamma(a+{\textstyle{1\over 2}})}+1$	(31)
$\quad {}_2F_1(a,b;c;{\textstyle{1\over 2}}) =2^a \,{}_2F_1(a,c-b;c;-1)$	(32)
$\quad {}_2F_1(a,b;{\textstyle{1\over 2}}(a+b+1);{\textstyle{1\over 2}}) = {\Gam... ...b)]\over\Gamma[{\textstyle{1\over 2}}(1+a)]\Gamma[{\textstyle{1\over 2}}(1+b)]}$	(33)
$\quad {}_2F_1(a,1-a;c;{\textstyle{1\over 2}}) = {\Gamma({\textstyle{1\over 2}}c... ...]\over\Gamma[{\textstyle{1\over 2}}(a+c)]\Gamma[{\textstyle{1\over 2}}(1+c-a)]}$	(34)
$\quad {}_2F_1(a,b;c;1) = {\Gamma(c)\Gamma(c-a-b)\over\Gamma(c-a)\Gamma(c-b)}.$	(35)

Kummer's First Formula gives

$${}_2F_1({\textstyle{1\over 2}}+m-k,-n;2m+1;1) = {\Gamma(2m+1... ...}+k+n)\over \Gamma(m+{\textstyle{1\over 2}}+k)\Gamma(2m+1+n)},$$ (36)

where $m\not= -1/2$, $-1$, $-3/2$, .... Many additional identities are given by Abramowitz and Stegun (1972, p. 557).

Hypergeometric functions can be generalized to Generalized Hypergeometric Functions

$${}_nF_m(a_1,\ldots,a_n; b_1,\ldots,b_m; z).$$ (37)

A function of the form ${}_1F_1(a;b;z)$ is called a Confluent Hypergeometric Function, and a function of the form ${}_0F_1(;b;z)$ is called a Confluent Hypergeometric Limit Function.

See also Appell Hypergeometric Function,
Barnes' Lemma, Bradley's Theorem, Cayley's Hypergeometric Function Theorem, Clausen Formula, Closed Form, Confluent Hypergeometric Function, Confluent Hypergeometric Limit Function, Contiguous Function, Darling's Products, Generalized Hypergeometric Function, Gosper's Algorithm, Hypergeometric Identity, Hypergeometric Series, Jacobi Polynomial, Kummer's Formulas, Kummer's Quadratic Transformation, Kummer's Relation, Orr's Theorem, Ramanujan's Hypergeometric Identity, Saalschützian, Sister Celine's Method, Zeilberger's Algorithm

References

Hypergeometric Functions

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Arfken, G. ``Hypergeometric Functions.'' §13.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 748-752, 1985.

Fine, N. J. Basic Hypergeometric Series and Applications. Providence, RI: Amer. Math. Soc., 1988.

Gasper, G. and Rahman, M. Basic Hypergeometric Series. Cambridge, England: Cambridge University Press, 1990.

Gauss, C. F. ``Disquisitiones Generales Circa Seriem Infinitam $\left[{\alpha\beta\over 1\cdot\gamma}\right]x+\left[{\alpha(\alpha+1)\beta(\beta+1)\over 1\cdot 2\cdot\gamma(\gamma+1)}\right]x^2$
$+\left[{\alpha(\alpha+1)(\alpha+2)\beta(\beta+1)(\beta+2)\over 1\cdot 2\cdot 3\cdot\gamma(\gamma+1)(\gamma+2)}\right]x^3+$ etc. Pars Prior.'' Commentationes Societiones Regiae Scientiarum Gottingensis Recentiores, Vol. II. 1813.

Gessel, I. and Stanton, D. ``Strange Evaluations of Hypergeometric Series.'' SIAM J. Math. Anal. 13, 295-308, 1982.

Gosper, R. W. ``Decision Procedures for Indefinite Hypergeometric Summation.'' Proc. Nat. Acad. Sci. USA 75, 40-42, 1978.

Goursat, M. E. ``Sur l'équation différentielle linéaire qui admet pour intégrale la série hypergéométrique.'' Ann. Sci. École Norm. Super. Sup. 10, S3-S142, 1881.

Iyanaga, S. and Kawada, Y. (Eds.). ``Hypergeometric Functions and Spherical Functions.'' Appendix A, Table 18 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1460-1468, 1980.

Kummer, E. E. ``Über die Hypergeometrische Reihe.'' J. für die Reine Angew. Mathematik 15, 39-83 and 127-172, 1837.

Magnus, W. and Oberhettinger, F. Formulas and Theorems for the Special Functions of Mathematical Physics. New York: Chelsea, 1949.

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 541-547, 1953.

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A. K. Peters, 1996.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Hypergeometric Functions.'' §6.12 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 263-265, 1992.

Seaborn, J. B. Hypergeometric Functions and Their Applications. New York: Springer-Verlag, 1991.

Snow, C. Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory. Washington, DC: U. S. Government Printing Office, 1952.

Spanier, J. and Oldham, K. B. ``The Gauss Function $F(a,b;c;x)$.'' Ch. 60 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 599-607, 1987.