A product involving an infinite number of terms. Such products can converge. In fact, for positive 
,
converges to a nonzero number iff 
converges.
Infinite products can be used to define the cosine
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(1) |
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(2) |
sine, and sinc function. They also appear in polygon circumscribing,
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(3) |
An interesting infinite product formula due to Euler which relates 
and the nth prime 
is
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(4) |
|
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(5) |
(Blatner 1997). Knar's formula gives a functional equation for the gamma function
in terms of the infinite product
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(6) |
A regularized product identity is given by
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(7) |
(Muñoz Garcia and Pérez-Marco 2003).
Mellin's formula states
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(8) |
where 
is the digamma function and is the gamma function.
The following class of products
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(9) |
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(10) |
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|
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(11) |
|
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(12) | |
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(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 r > 1,
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(14) |
where 
(Borwein et al. 20034, 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).
The first few products
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(15) |
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(16) |
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(17) |
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|
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(18) |
where
,
,
are the roots of
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(19) |
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(20) |
and
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(21) |
respectively, can also be done analytically. Note that (20) and (21) were unknown to Borwein and Corless (1999).
The product
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(22) |
has closed form expressions for small positive integral 
,
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(23) |
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|
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(24) |
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(25) |
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(26) |
The d-analog expression
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(27) |
also has closed form expressions,
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|
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(28) |
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(29) |
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(30) |
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(31) |
General expressions for infinite products of this type include
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(32) |
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(33) |
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(34) |
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|
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(35) |
where
is the gamma function and 
denotes the complex modulus (Kahovec). (32) and (33) can also be rewritten as
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(36) | |
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(37) |
where
is the floor function, 
is the ceiling function, and 
is the modulus of a (mod m) (Kahovec).
Infinite products of the form
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(38) |
converge for n > 1, where 
is a Jacobi elliptic function. The first few such products are numerically given by
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(39) |
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|
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(40) |
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(41) |
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(42) |
(Sloane's A048651, A100220, A100221, and A100222).
The following analogous classes of products can also be done analytically (J. Zúñiga, pers. comm., Nov. 9, 2004), where agin is a Jacobi elliptic function,
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(43) |
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(44) |
|
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(45) | |
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(46) |
|
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(47) | |
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(48) |
|
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(49) | |
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(50) |
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(51) |
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(52) |
|
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(53) |
A class of infinite products derived from the Barnes G-function is given by
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(54) |
where 
is the Euler-Mascheroni constant. For z = 1, 2, 3, and 4, the explicit products are given by
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|
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(55) |
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(56) |
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(57) |
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(58) |
The interesting identities
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(59) |
(Ewell 1995, 1999), where b(n) is the exponent of the exact power of 2 dividing n, 
is the odd part of n, 
is the divisor function of n, and 
is the sum of squares function, and
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(60) |
(Ewell 1998, 1999) arise is connection with the tau function.
An unexpected infinite product involving 
is given by
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(61) |
(Dobinski 1876, Agnew and Walker 1947).
A curious identity first noted by Gosper is given by
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|
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(62) |
|
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(63) |
(Sloane's A100072), where
is the gamma function, 
is the trigamma function, and A is the Glaisher-Kinkelin constant.
Artin's Constant, Barnes G-Function, Cosine, d-Analog, Dedekind Eta Function, Dirichlet Eta Function, Dobinski's Formula, Euler Identity, Euler-Mascheroni Constant, Euler Product, Euler's Pentagonal Number Theorem, Euler Product, Gamma Function, Hadamard Product, Jacobi Triple Product, Knar's Formula, Mellin's Formula, Mertens Theorem, Polygon Circumscribing, Polygon Inscribing, Power Tower, Prime Products, Q-Function, q-Series, Riemann Zeta Function, Sine, Stephens' Constant, Wallis Formula
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. Natick, 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, 1999.
Finch, S. R. "Kepler-Bouwkamp Constant." §6.3 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 428-429, 2003.
Hansen, E. R. A Table of Series and Products. Englewood Cliffs, NJ: Prentice-Hall, 1975.
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.
Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, 2003.
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 Garcia, E. and Pérez-Marco, R. "The Product Over All Primes is 
.
Ritt, J. F. "Representation of Analytic Functions as Infinite Products." Math. Z. 32, 1-3, 1930.
Sloane, N. J. A. Sequences A048651, A100072, A100220, A100221, and A100222 in "The On-Line Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/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.