Infinite Product

An infinite product is 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 García and Pérez Marco 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,

(14)

where and are positive integers, , and (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)

where and are positive integers, , and . The and are the roots, counted with multiplicity, of the polynomials and , respectively. If no or is a positive integer, the product converges to the stated nonzero value (P. Abbott, pers. comm., Mar. 30, 2006; National Institute of Standards and Technology, §5.8).

For integers ,

(20)

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

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

For integers and , these are a special case of the general formula

(27)

(Prudnikov et al. 1986, p. 754).

Similarly, for integers ,

(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)

For a positive integer and nonzero real number , general expressions for infinite products of this type include

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

where is the gamma function and denotes the complex modulus (Kahovec). Values at poles of the gamma function are understood by limits. The even-degree products can also be written 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 theta function. Here, the case is exactly the constant encountered in the analysis of digital tree searching.

Other products include

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

(OEIS A086056 and A247559; Prudnikov et al. 1986, p. 757). The first expression is a limit of even partial products. The ordinary infinite product does not converge. Note that Prudnikov et al. (1986, p. 757) also incorrectly give the product

(52)

where is a q-Pochhammer symbol, as , which differs from the correct result by .

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

			(53)
			(54)
			(55)
			(56)
			(57)
			(58)
			(59)
			(60)
			(61)
			(62)
			(63)

For a real number and , the final alternating product has positive factors and converges to the stated hypergeometric function and gamma function expressions (National Institute of Standards and Technology, §5.8 and §15.4). The first geometric product 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

(64)

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

			(65)
			(66)
			(67)
			(68)

The identities

(69)

(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

			(70)
			(71)

(OEIS 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 in connection with the tau function.

For a real number such that every factor is defined and nonzero, and , an infinite product involving the tangent is given by

(72)

(Dobinski 1876, Agnew and Walker 1947).

An identity first noted by Gosper is given by

			(73)
			(74)

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