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Gamma Function

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GammaFunction

The (complete) gamma functionGamma(n) is defined to be an extension of thefactorial tocomplex andreal number arguments. It is related to thefactorial by

Gamma(n)=(n-1)!,
(1)

a slightly unfortunate notation due to Legendre which is now universally used instead of Gauss's simplerPi(n)=n! (Gauss 1812; Edwards 2001, p. 8).

It isanalytic everywhere except atz=0,-1,-2, ..., and the residue atz=-k is

Res_(z=-k)Gamma(z)=((-1)^k)/(k!).
(2)

There are no pointsz at whichGamma(z)=0.

The gamma function is implemented in theWolframLanguage asGamma[z].

There are a number of notational conventions in common use for indication of a power of a gamma functions. While authors such as Watson (1939) useGamma^n(z) (i.e., using a trigonometric function-like convention), it is also common to write[Gamma(z)]^n.

The gamma function can be defined as adefinite integral forR[z]>0 (Euler's integral form)

Gamma(z)=int_0^inftyt^(z-1)e^(-t)dt
(3)
=2int_0^inftye^(-t^2)t^(2z-1)dt,
(4)

or

Gamma(z)=int_0^1[ln(1/t)]^(z-1)dt.
(5)

The complete gamma functionGamma(x) can be generalized to the upperincomplete gamma functionGamma(a,x) and lowerincomplete gamma functiongamma(a,x).

GammaReIm
GammaContours

Plots of the real and imaginary parts ofGamma(z) in the complex plane are illustrated above.

Integrating equation (3)by parts for areal argument, it can be seen that

Gamma(x)=int_0^inftyt^(x-1)e^(-t)dt
(6)
=[-t^(x-1)e^(-t)]_0^infty+int_0^infty(x-1)t^(x-2)e^(-t)dt
(7)
=(x-1)int_0^inftyt^(x-2)e^(-t)dt
(8)
=(x-1)Gamma(x-1).
(9)

Ifx is anintegern=1, 2, 3, ..., then

Gamma(n)=(n-1)Gamma(n-1)
(10)
=(n-1)(n-2)Gamma(n-2)
(11)
=(n-1)(n-2)...1
(12)
=(n-1)!,
(13)

so the gamma function reduces to thefactorial forapositive integer argument.

A beautiful relationship betweenGamma(z) and theRiemann zeta functionzeta(z) is given by

zeta(z)Gamma(z)=int_0^infty(u^(z-1))/(e^u-1)du
(14)

forR[z]>1 (Havil 2003, p. 60).

The gamma function can also be defined by aninfiniteproduct form (Weierstrass form)

Gamma(z)=[ze^(gammaz)product_(r=1)^infty(1+z/r)e^(-z/r)]^(-1),
(15)

wheregamma is theEuler-Mascheroni constant (Krantz 1999, p. 157; Havil 2003, p. 57). Taking the logarithm of both sides of (◇),

-ln[Gamma(z)]=lnz+gammaz+sum_(n=1)^infty[ln(1+z/n)-z/n].
(16)

Differentiating,

-(Gamma^'(z))/(Gamma(z))=1/z+gamma+sum_(n=1)^(infty)((1/n)/(1+z/n)-1/n)
(17)
=1/z+gamma+sum_(n=1)^(infty)(1/(n+z)-1/n)
(18)
Gamma^'(z)=-Gamma(z)[1/z+gamma+sum_(n=1)^(infty)(1/(n+z)-1/n)]
(19)
=Gamma(z)Psi(z)
(20)
=Gamma(z)psi_0(z)
(21)
Gamma^'(1)=-Gamma(1){1+gamma+[(1/2-1)+(1/3-1/2)+...+(1/(n+1)-1/n)+...]}
(22)
=-(1+gamma-1)
(23)
=-gamma
(24)
Gamma^'(n)=-Gamma(n){1/n+gamma+[(1/(1+n)-1)+(1/(2+n)-1/2)+(1/(3+n)-1/3)+...]}
(25)
=-(n-1)!(1/n+gamma-sum_(k=1)^(n)1/k),
(26)

wherePsi(z) is thedigamma function andpsi_0(z) is thepolygamma function.nth derivatives are given in terms of thepolygamma functionspsi_n,psi_(n-1), ...,psi_0.

The minimum valuex_0 ofGamma(x) forrealpositivex=x_0 is achieved when

Gamma^'(x_0)=Gamma(x_0)psi_0(x_0)=0
(27)
psi_0(x_0)=0.
(28)

This can be solved numerically to givex_0=1.46163... (OEISA030169; Wrench 1968), which hascontinued fraction [1, 2, 6, 63, 135, 1, 1, 1, 1, 4, 1, 38, ...] (OEISA030170). Atx_0,Gamma(x_0) achieves the value 0.8856031944... (OEISA030171), which hascontinued fraction [0, 1, 7, 1, 2, 1, 6, 1, 1, ...] (OEISA030172).

The Euler limit form is

Gamma(z)=1/zproduct_(n=1)^infty[(1+1/n)^z(1+z/n)^(-1)],
(29)

so

Gamma(z)=lim_(n->infty)((n+1)^z)/(z(1+z)(1+z/2)(1+z/3)...(1+z/n))
(30)
=lim_(n->infty)((n+1)^zn!)/(z(z+1)(z+2)(z+3)...(z+n))
(31)
=lim_(n->infty)(n!)/((z)_(n+1))(n+1)^z
(32)
=lim_(n->infty)(n!)/((z)_(n+1))n^z
(33)

(Krantz 1999, p. 156).

One over the gamma function1/Gamma(z) is anentire function and can be expressed as

1/(Gamma(z))=zexp[gammaz-sum_(k=2)^infty((-1)^kzeta(k)z^k)/k],
(34)

wheregamma is theEuler-Mascheroni constant andzeta(z) is theRiemann zeta function (Wrench 1968). Anasymptotic series for1/Gamma(z) is given by

1/(Gamma(z))∼z+gammaz^2+1/(12)(6gamma^2-pi^2)z^3+1/(12)[2gamma^3-gammapi^2+4zeta(3)]z^4+....
(35)

Writing

1/(Gamma(z))=sum_(k=1)^inftya_kz^k,
(36)

thea_k satisfy

a_n=na_1a_n-a_2a_(n-1)+sum_(k=2)^n(-1)^kzeta(k)a_(n-k)
(37)

(Bourguet 1883, Davis 1933, Isaacson and Salzer 1943, Wrench 1968). Wrench (1968) numerically computed the coefficients for the series expansion about 0 of

1/(z(1+z)Gamma(z))=1+(gamma-1)z+[1+1/2(gamma-2)gamma-1/(12)pi^2]z^2+....
(38)

TheLanczos approximation gives a series expansion forGamma(z+1) forz>0 in terms of an arbitrary constantsigma such thatR[z+sigma+1/2]>0.

The gamma function satisfies thefunctional equations

Gamma(1+z)=zGamma(z)
(39)
Gamma(1-z)=-zGamma(-z).
(40)

Additional identities are

Gamma(x)Gamma(-x)=-pi/(xsin(pix))
(41)
Gamma(x)Gamma(1-x)=pi/(sin(pix))
(42)
|(ix)!|^2=(pix)/(sinh(pix))
(43)
|(n+ix)!|=sqrt((pix)/(sinh(pix)))product_(s=1)^(n)sqrt(s^2+x^2).
(44)

Using (41), the gamma functionGamma(r) of a rational numberr can be reduced to a constant timesGamma(frac(r)) or1/Gamma(frac(r)). For example,

Gamma(2/3)=(2pi)/(sqrt(3)Gamma(1/3))
(45)
Gamma(3/4)=(sqrt(2)pi)/(Gamma(1/4))
(46)
Gamma(3/5)=sqrt(2-2/(sqrt(5)))pi/(Gamma(2/5))
(47)
Gamma(4/5)=sqrt(2+2/(sqrt(5)))pi/(Gamma(1/5)).
(48)

ForR[z]=-1/2,

|(-1/2+iy)!|^2=pi/(cosh(piy)).
(49)

Gamma functions of argument2z can be expressed using theLegendre duplication formula

Gamma(2z)=(2pi)^(-1/2)2^(2z-1/2)Gamma(z)Gamma(z+1/2).
(50)

Gamma functions of argument3z can be expressed using a triplication formula

Gamma(3z)=(2pi)^(-1)3^(3z-1/2)Gamma(z)Gamma(z+1/3)Gamma(z+2/3).
(51)

The general result is theGauss multiplicationformula

Gamma(z)Gamma(z+1/n)...Gamma(z+(n-1)/n)=(2pi)^((n-1)/2)n^(1/2-nz)Gamma(nz).
(52)

The gamma function is also related to theRiemann zeta functionzeta(z) by

Gamma(s/2)pi^(-s/2)zeta(s)=Gamma((1-s)/2)pi^(-(1-s)/2)zeta(1-s).
(53)

For integern=1, 2, ..., the first few values ofGamma(n) are 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, ... (OEISA000142). For half-integer arguments,Gamma(n/2) has the special form

Gamma(1/2n)=((n-2)!!sqrt(pi))/(2^((n-1)/2)),
(54)

wheren!! is adouble factorial. The first few values forn=1, 3, 5, ... are therefore

Gamma(1/2)=sqrt(pi)
(55)
Gamma(3/2)=1/2sqrt(pi)
(56)
Gamma(5/2)=3/4sqrt(pi),
(57)

15sqrt(pi)/8,105sqrt(pi)/16, ... (OEISA001147 andA000079; Wells 1986, p. 40). In general, forn apositive integern=1, 2, ...

Gamma(1/2+n)=(1·3·5...(2n-1))/(2^n)sqrt(pi)
(58)
=((2n-1)!!)/(2^n)sqrt(pi)
(59)
Gamma(1/2-n)=((-1)^n2^n)/(1·3·5...(2n-1))sqrt(pi)
(60)
=((-1)^n2^n)/((2n-1)!!)sqrt(pi).
(61)

Simple closed-form expressions of this type do not appear to exist forGamma(1/n) forn a positive integern>2. However, Borwein and Zucker (1992) give a variety of identities relating gamma functions to square roots andelliptic integral singular valuesk_n, i.e.,elliptic modulik_n such that

(K^'(k_n))/(K(k_n))=sqrt(n),
(62)

whereK(k) is acomplete elliptic integral of the first kind andK^'(k)=K(k^')=K(sqrt(1-k^2)) is the complementary integral. M. Trott (pers. comm.) has developed an algorithm for automatically generating hundreds of such identities.

Gamma(1/3)=2^(7/9)3^(-1/12)pi^(1/3)[K(k_3)]^(1/3)
(63)
Gamma(1/4)=2pi^(1/4)[K(k_1)]^(1/2)
(64)
Gamma(1/6)=2^(-1/3)3^(1/2)pi^(-1/2)[Gamma(1/3)]^2
(65)
Gamma(1/8)Gamma(3/8)=(sqrt(2)-1)^(1/2)2^(13/4)pi^(1/2)K(k_2)
(66)
(Gamma(1/8))/(Gamma(3/8))=2(sqrt(2)+1)^(1/2)pi^(-1/4)[K(k_1)]^(1/2)
(67)
Gamma(1/(12))=2^(-1/4)3^(3/8)(sqrt(3)+1)^(1/2)pi^(-1/2)Gamma(1/4)Gamma(1/3)
(68)
Gamma(5/(12))=2^(1/4)3^(-1/8)(sqrt(3)-1)^(1/2)pi^(1/2)(Gamma(1/4))/(Gamma(1/3))
(69)
(Gamma(1/(24))Gamma((11)/(24)))/(Gamma(5/(24))Gamma(7/(24)))=sqrt(3)sqrt(2+sqrt(3))
(70)
(Gamma(1/(24))Gamma(5/(24)))/(Gamma(7/(24))Gamma((11)/(24)))=4·3^(1/4)(sqrt(3)+sqrt(2))pi^(-1/2)K(k_1)
(71)
(Gamma(1/(24))Gamma(7/(24)))/(Gamma(5/(24))Gamma((11)/(24)))=2^(25/18)3^(1/3)(sqrt(2)+1)pi^(-1/3)[K(k_3)]^(2/3)
(72)
Gamma(1/(24))Gamma(5/(24))Gamma(7/(24))Gamma((11)/(24))=384(sqrt(2)+1)(sqrt(3)-sqrt(2))(2-sqrt(3))pi[K(k_6)]^2
(73)
Gamma(1/(10))=2^(-7/10)5^(1/4)(sqrt(5)+1)^(1/2)pi^(-1/2)Gamma(1/5)Gamma(2/5)
(74)
Gamma(3/(10))=2^(-3/5)(sqrt(5)-1)pi^(1/2)(Gamma(1/5))/(Gamma(2/5))
(75)
(Gamma(1/(15))Gamma(4/(15))Gamma(7/(15)))/(Gamma(2/(15)))=2·3^(1/2)5^(1/6)sin(2/(15)pi)[Gamma(1/3)]^2
(76)
(Gamma(1/(15))Gamma(2/(15))Gamma(7/(15)))/(Gamma(4/(15)))=2^2·3^(2/5)sin(1/5pi)sin(4/(15)pi)[Gamma(1/5)]^2
(77)
(Gamma(2/(15))Gamma(4/(15))Gamma(7/(15)))/(Gamma(1/(15)))=(2^(-3/2)3^(-1/5)5^(1/4)(sqrt(5)-1)^(1/2)[Gamma(2/5)]^2)/(sin(4/(15)pi))
(78)
(Gamma(1/(15))Gamma(2/(15))Gamma(4/(15)))/(Gamma(7/(15)))=60(sqrt(5)-1)sin(7/(15)pi)[K(k_(15))]^2
(79)
(Gamma(1/(20))Gamma(9/(20)))/(Gamma(3/(20))Gamma(7/(20)))=2^(-1)5^(1/4)(sqrt(5)+1)
(80)
(Gamma(1/(20))Gamma(3/(20)))/(Gamma(7/(20))Gamma(9/(20)))=2^(4/5)(10-2sqrt(5))^(1/2)pi^(-1)sin(7/(20)pi)sin(9/(20)pi)[Gamma(1/5)]^2
(81)
(Gamma(1/(20))Gamma(7/(20)))/(Gamma(3/(20))Gamma(9/(20)))=2^(3/5)(10+2sqrt(5))^(1/2)pi^(-1)sin(3/(20)pi)sin(9/(20)pi)[Gamma(2/5)]^2
(82)
Gamma(1/(20))Gamma(3/(20))Gamma(7/(20))Gamma(9/(20))=160(sqrt(5)-2)^(1/2)pi[K(k_5)]^2.
(83)

Several of these are also given in Campbell (1966, p. 31).

A few curious identities include

product_(n=1)^(2)Gamma(1/3n)=(2pi)/(sqrt(3))
(84)
product_(n=1)^(3)Gamma(1/3n)=(2pi)/(sqrt(3))
(85)
product_(n=1)^(4)Gamma(1/3n)=(2piGamma(1/3))/(3sqrt(3))
(86)
product_(n=1)^(5)Gamma(1/3n)=8/(27)pi^2
(87)
product_(n=1)^(6)Gamma(1/3n)=8/(27)pi^2
(88)
product_(n=1)^(7)Gamma(1/3n)=(32)/(243)pi^2Gamma(1/3)
(89)
product_(n=1)^(8)Gamma(1/3n)=(640pi^3)/(2187sqrt(3)),
(90)

of which Magnus and Oberhettinger (1949, p. 1) give only the last case and

(Gamma^'(1))/(Gamma(1))-(Gamma^'(1/2))/(Gamma(1/2))=2ln2
(91)

(Magnus and Oberhettinger 1949, p. 1). Ramanujan also gave a number of fascinating identities:

(Gamma^2(n+1))/(Gamma(n+xi+1)Gamma(n-xi+1))=product_(k=1)^infty[1+(x^2)/((n+k)^2)]
(92)
phi(m,n)phi(n,m)=(Gamma^3(m+1)Gamma^3(n+1))/(Gamma(2m+n+1)Gamma(2n+m+1)) ×(cosh[pi(m+n)sqrt(3)]-cos[pi(m-n)])/(2pi^2(m^2+mn+n^2)),
(93)

where

phi(m,n)=product_(k=1)^infty[1+((m+n)/(k+m))^3],
(94)
product_(k=1)^infty[1+(n/k)^3]product_(k=1)^infty[1+3(n/(n+2k))^2]=(Gamma(1/2n))/(Gamma[1/2(n+1)])(cosh(pinsqrt(3))-cos(pin))/(2^(n+2)pi^(3/2)n)
(95)

(Berndt 1994).

Ramanujan gave the infinite sums

sum_(k=0)^infty(8k+1)[(Gamma(k+1/4))/(k!Gamma(1/4))]^4 =1+9(1/4)^4+17((1·5)/(4·8))^4+25((1·5·9)/(4·8·12))^4+... =(2^(3/2))/(sqrt(pi)[Gamma(3/4)]^2)
(96)

and

sum_(k=0)^infty(-1)^k(4k+1)[((2k-1)!!)/((2k)!!)]^5 =1-5(1/2)^5+9((1·3)/(2·4))^5-13((1·3·5)/(2·4·6))^5+... =2/([Gamma(3/4)]^4)
(97)

(Hardy 1923; Hardy 1924; Whipple 1926; Watson 1931; Bailey 1935; Hardy 1999, p. 7).

The followingasymptotic series is occasionally useful in probability theory (e.g., theone-dimensional random walk):

(Gamma(J+1/2))/(Gamma(J))=sqrt(J)(1-1/(8J)+1/(128J^2)+5/(1024J^3)-(21)/(32768J^4)+...)
(98)

(OEISA143503 andA061549; Grahamet al.1994). This series also gives a nice asymptotic generalization ofStirling numbers of the first kind to fractional values.

It has long been known thatGamma(1/4)pi^(-1/4) istranscendental (Davis 1959), as isGamma(1/3) (Le Lionnais 1983; Borwein and Bailey 2003, p. 138), and Chudnovsky has apparently recently proved thatGamma(1/4) is itselftranscendental (Borwein and Bailey 2003, p. 138).

There exist efficient iterative algorithms forGamma(k/24) for all integersk (Borwein and Bailey 2003, p. 137). For example, a quadratically converging iteration forGamma(1/4)=3.6256099... (OEISA068466) is given by defining

x_n=1/2(x_(n-1)^(1/2)+x_(n-1)^(-1/2))
(99)
y_n=(y_(n-1)x_(n-1)^(1/2)+x_(n-1)^(-1/2))/(y_(n-1)+1),
(100)

settingx_0=sqrt(2) andy_1=2^(1/4), and then

Gamma(1/4)=2(1+sqrt(2))^(3/4)[product_(n=1)^inftyx_n^(-1)((1+x_n)/(1+y_n))^3]^(1/4)
(101)

(Borwein and Bailey 2003, pp. 137-138).

No such iteration is known forGamma(1/5) (Borwein and Borwein 1987; Borwein and Zucker 1992; Borwein and Bailey 2003, p. 138).

See also

Bailey's Theorem,Barnes G-Function,Binet's Fibonacci Number Formula,Bohr-Mollerup Theorem,Digamma Function,Fransén-Robinson ConstantGauss Multiplication Formula,Incomplete Gamma Function,Knar's Formula,Lambda Function,Lanczos Approximation,Legendre Duplication Formula,Log Gamma Function,Mellin's Formula,Mu Function,Nu Function,Pearson's Function,Polygamma Function,Regularized Gamma Function,Stirling's Series,SuperfactorialExplore this topic in the MathWorld classroom

Related Wolfram sites

http://functions.wolfram.com/GammaBetaErf/Gamma/,http://functions.wolfram.com/GammaBetaErf/LogGamma/

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References

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Wellesley, MA: A K Peters, 2003.Borwein, J. and Borwein, P. B.Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, p. 6, 1987.Borwein, J. M. and Zucker, I. J. "Fast Evaluation of the Gamma Function for Small Rational Fractions Using Complete Elliptic Integrals of the First Kind."IMA J. Numerical Analysis12, 519-526, 1992.Bourguet, L. "Sur les intégrales Eulériennes et quelques autres fonctions uniformes."Acta Math.2, 261-295, 1883.Campbell, R.Les intégrales eulériennes et leurs applications. Paris: Dunod, 1966.Davis, H. T.Tables of the Higher Mathematical Functions. Bloomington, IN: Principia Press, 1933.Davis, P. J. "Leonhard Euler's Integral: A Historical Profile of the Gamma Function."Amer. Math. Monthly66, 849-869, 1959.Edwards, H. M.Riemann's Zeta Function. New York: Dover, 2001.Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. "The Gamma Function." Ch. 1 inHigher Transcendental Functions, Vol. 1. New York: Krieger, pp. 1-55, 1981.Finch, S. R. "Euler-Mascheroni Constant." §1.5 inMathematical Constants. Cambridge, England: Cambridge University Press, pp. 28-40, 2003.Gauss, C. F. "Disquisitiones Generales Circa Seriem Infinitam[(alphabeta)/(1·gamma)]x+[(alpha(alpha+1)beta(beta+1))/(1·2·gamma(gamma+1))]x^2+[(alpha(alpha+1)(alpha+2)beta(beta+1)(beta+2))/(1·2·3·gamma(gamma+1)(gamma+2))]x^3+ etc. Pars Prior."Commentationes Societiones Regiae Scientiarum Gottingensis Recentiores, Vol. II. 1812. Reprinted inGesammelte Werke, Bd. 3, pp. 123-163 and 207-229, 1866.Graham, R. L.; Knuth, D. E.; and Patashnik, O. Answer to Problem 9.60 inConcrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994.Hardy, G. H. "A Chapter from Ramanujan's Note-Book."Proc. Cambridge Philos. Soc.21, 492-503, 1923.Hardy, G. H. "Some Formulae of Ramanujan."Proc. London Math. Soc. (Records of Proceedings at Meetings)22, xii-xiii, 1924.Hardy, G. H.Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.Havil, J. "The Gamma Function." Ch. 6 inGamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 53-60, 2003.Isaacson, E. and Salzer, H. E. "Mathematical Tables--Errata: 19. J. P. L. Bourget, 'Sur les intégrales Eulériennes et quelques autres fonctions uniformes,'Acta Mathematica, v. 2, 1883, pp. 261-295.' "Math. Tab. Aids Comput.1, 124, 1943.Koepf, W. "The Gamma Function." Ch. 1 inHypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 4-10, 1998.Krantz, S. G. "The Gamma and Beta Functions." §13.1 inHandbook of Complex Variables. Boston, MA: Birkhäuser, pp. 155-158, 1999.Le Lionnais, F.Les nombres remarquables. Paris: Hermann, p. 46, 1983.Magnus, W. and Oberhettinger, F.Formulas and Theorems for the Special Functions of Mathematical Physics. New York: Chelsea, 1949.Nielsen, N. "Handbuch der Theorie der Gammafunktion." Part I inDie Gammafunktion. New York: Chelsea, 1965.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Gamma Function, Beta Function, Factorials, Binomial Coefficients" and "Incomplete Gamma Function, Error Function, Chi-Square Probability Function, Cumulative Poisson Function." §6.1 and 6.2 inNumerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 206-209 and 209-214, 1992.Sloane, N. J. A. SequencesA000079/M1129,A000142/M1675,A001147/M3002,A030169,A030170,A030171,A030172,A061549,A068466, andA143503 in "The On-Line Encyclopedia of Integer Sequences."Spanier, J. and Oldham, K. B. "The Gamma FunctionGamma(x)" and "The Incomplete Gammagamma(nu;x) and Related Functions." Chs. 43 and 45 inAn Atlas of Functions. Washington, DC: Hemisphere, pp. 411-421 and 435-443, 1987.Watson, G. N. "Theorems Stated by Ramanujan (XI)."J. London Math. Soc.6, 59-65, 1931.Watson, G. N. "Three Triple Integrals."Quart. J. Math., Oxford Ser. 210, 266-276, 1939.Wells, D.The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 40, 1986.Whipple, F. J. W. "A Fundamental Relation between Generalised Hypergeometric Series."J. London Math. Soc.1, 138-145, 1926.Whittaker, E. T. and Watson, G. N.A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.Wrench, J. W. Jr. "Concerning Two Series for the Gamma Function."Math. Comput.22, 617-626, 1968.

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Gamma Function

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Weisstein, Eric W. "Gamma Function." FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/GammaFunction.html

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