A product involving an infinite number of terms. Such products can converge. In fact, for positive , the product converges to a nonzero number iff converges.

Infinite products can be used to define the cosine

(1)

gamma function

(2)

sine, and sinc function. They also appear in polygon circumscribing,

(3)

An interesting infinite product formula due to Euler which relates and the th prime is

			(4)
			(5)

(Blatner 1997). Knar's formula gives a functional equation for the gamma function in terms of the infinite product

(6)

A regularized product identity is given by

(7)

(Muñoz Garcia and Pérez-Marco 2003, 2008).

Mellin's formula states

(8)

where is the digamma function and is the gamma function.

The following class of products

			(9)
			(10)
			(11)
			(12)
			(13)

(Borwein et al. 2004, pp. 4-6), where is the gamma function, the first of which is given in Borwein and Corless (1999), can be done analytically. In particular, for ,

(14)

where (Borwein et al. 2004, pp. 6-7). It is not known if (13) is algebraic, although it is known to satisfy no integer polynomial with degree less than 21 and Euclidean norm less than (Borwein et al. 2004, p. 7).

Products of the following form can be done analytically,

(15)

where , , and are the roots of

			(16)
			(17)
			(18)

respectively, can also be done analytically. Note that (17) and (18) were unknown to Borwein and Corless (1999). These are special cases of the result that

(19)

if and , where is the th root of and is the th root of (P. Abbott, pers. comm., Mar. 30, 2006).

For ,

(20)

(D. W. Cantrell, pers. comm., Apr. 18, 2006). The first few explicit cases are

			(21)
			(22)
			(23)
			(24)
			(25)
			(26)

These are a special case of the general formula

(27)

(Prudnikov et al. 1986, p. 754).

Similarly, for ,

(28)

(D. W. Cantrell, pers. comm., Mar. 29, 2006). The first few explicit cases are

			(29)
			(30)
			(31)
			(32)
			(33)
			(34)

The d-analog expression

(35)

also has closed form expressions,

			(36)
			(37)
			(38)
			(39)

General expressions for infinite products of this type include

			(40)
			(41)
			(42)
			(43)

where is the gamma function and denotes the complex modulus (Kahovec). (40) and (41) can also be rewritten as

			(44)
			(45)

where is the floor function, is the ceiling function, and is the modulus of (mod ) (Kahovec).

Infinite products of the form

			(46)
			(47)

converge for , where is a q-Pochhammer symbol and is a Jacobi elliptic function. Here, the case is exactly the constant encountered in the analysis of digital tree searching.

Other products include

					(48)
					(49)
					(50)
					(51)

(Sloane's A086056 and A118254; Prudnikov et al. 1986, p. 757).

The following analogous classes of products can also be done analytically (J. Zúñiga, pers. comm., Nov. 9, 2004), where again is a Jacobi elliptic function,

			(52)
			(53)
			(54)
			(55)
			(56)
			(57)
			(58)
			(59)
			(60)
			(61)
			(62)

The first of these can be used to express the Fibonacci factorial constant in closed form.

A class of infinite products derived from the Barnes G-function is given by

(63)

where is the Euler-Mascheroni constant. For , 2, 3, and 4, the explicit products are given by

			(64)
			(65)
			(66)
			(67)

The interesting identities

(68)

(Ewell 1995, 2000), where is the exponent of the exact power of 2 dividing , is the odd part of , is the divisor function of , and

			(69)
			(70)

(Sloane's A101127; Jacobi 1829; Ford et al. 1994; Ewell 1998, 2000), the latter of which is known as "aequatio identica satis abstrusa" in the string theory physics literature, arise is connection with the tau function.

An unexpected infinite product involving is given by

(71)

(Dobinski 1876, Agnew and Walker 1947).

A curious identity first noted by Gosper is given by

			(72)
			(73)

(Sloane's A100072), where is the gamma function, is the trigamma function, and is the Glaisher-Kinkelin constant.

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 75, 1972.

Agnew, R. P. and Walker, R. J. "A Trigonometric Infinite Product." Amer. Math. Monthly 54, 206-211, 1947.

Arfken, G. "Infinite Products." §5.11 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 346-351, 1985.

Blatner, D. The Joy of Pi. New York: Walker, p. 119, 1997.

Borwein, J.; Bailey, D.; and Girgensohn, R. "Two Products." §1.2 in Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, pp. 4-7, 2004.

Borwein, J. M. and Corless, R. M. "Emerging Tools for Experimental Mathematics." Amer. Math. Monthly 106, 899-909, 1999.

Dobinski, G. "Product einer unendlichen Factorenreihe." Archiv Math. u. Phys. 59, 98-100, 1876.

Erdélyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 1. New York: Krieger, p. 6, 1981.

Ewell, J. A. "Arithmetical Consequences of a Sextuple Product Identity." Rocky Mtn. J. Math. 25, 1287-1293, 1995.

Ewell, J. A. "A Note on a Jacobian Identity." Proc. Amer. Math. Soc. 126, 421-423, 1998.

Ewell, J. A. "New Representations of Ramanujan's Tau Function." Proc. Amer. Math. Soc. 128, 723-726, 2000.

Finch, S. R. "Kepler-Bouwkamp Constant." §6.3 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 428-429, 2003.

Ford, D.; McKay, J.; and Norton, S. P. "More on Replicable Functions." Commun. Alg. 22, 5175-5193, 1994.

Hansen, E. R. A Table of Series and Products. Englewood Cliffs, NJ: Prentice-Hall, 1975.

Jacobi, C. G. J. "E formulis (7.),(8.) sequitur aequatio identica satis abstrusa: (14.) ." Fundamenta Nova Theoriae Functionum Ellipticarum. Königsberg, Germany: Regiomonti, Sumtibus fratrum Borntraeger, 1829. Reprinted in Gesammelte Werke, Band. 1. Providence, RI: Amer. Math. Soc., p. 147, 1969.

Jeffreys, H. and Jeffreys, B. S. "Infinite Products." §1.14 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 52-53, 1988.

Krantz, S. G. "The Concept of an Infinite Product." §8.1.6 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 104-105, 1999.

Muñoz García, E. and Pérez Marco, R. "The Product Over All Primes is ." Preprint IHES/M/03/34. May 2003. http://inc.web.ihes.fr/prepub/PREPRINTS/M03/Resu/resu-M03-34.html.

Muñoz García, E. and Pérez Marco, R. "The Product Over All Primes is ." Commun. Math. Phys. 277, 69-81, 2008.

Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. "Infinite Products." §6.2 in Integrals and Series, Vol. 1: Elementary Functions. New York: Gordon & Breach, pp. 753-757, 1986.

Ritt, J. F. "Representation of Analytic Functions as Infinite Products." Math. Z. 32, 1-3, 1930.

Sloane, N. J. A. Sequences A048651, A086056, A100072, A100220, A100221, A100222, A101127, and A118254 in "The On-Line Encyclopedia of Integer Sequences."

Whittaker, E. T. and Watson, G. N. §7.5-7.6 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, 1990.

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Weisstein, Eric W. "Infinite Product." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/InfiniteProduct.html