The Jacobi triple product or the Jacobi triple product identity is an identity between infinite products and series that allows the theta function of Carl Gustav Jacobi to be represented as an infinite product instead of an infinite series.
A special case is the pentagonal number theorem by Leonhard Euler , on the Jacobi also evidence of identity is based (Jacobi, Fundamenta Nova theoriae Functionum Ellipticarum, 1829).
The triple product identity is (with complex numbers , and )
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{\ displaystyle x, y}
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{\ displaystyle | x | <1}
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{\ displaystyle y \ neq 0}
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{\ displaystyle \ prod _ {m = 1} ^ {\ infty} \ left (1-x ^ {2m} \ right) \ left (1 + x ^ {2m-1} y ^ {2} \ right) \ left (1 + x ^ {2m-1} y ^ {- 2} \ right) = \ sum _ {n = - \ infty} ^ {\ infty} x ^ {n ^ {2}} y ^ {2n} .}
This can also be expressed as a relationship between theta functions. Let (where is the imaginary part of ) and . Then the right hand side of the triple product identity is the Jacobian theta function:
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exp
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{\ displaystyle x = \ exp (i \ pi \ tau)}
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{\ displaystyle \ tau> 0}
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exp
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{\ displaystyle y = \ exp (i \ pi z)}
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exp
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{\ displaystyle \ vartheta (z; \ tau) = \ sum _ {n = - \ infty} ^ {\ infty} \ exp (i \ pi n ^ {2} \ tau + 2i \ pi nz)}
.
and you get in total:
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{\ displaystyle \ vartheta (z; \ tau) = \ prod _ {m = 1} ^ {\ infty} \ left (1-e ^ {2m \ pi {\ rm {i}} \ tau} \ right) \ left [1 + e ^ {(2m-1) \ pi {\ rm {i}} \ tau +2 \ pi {\ rm {i}} z} \ right] \ left [1 + e ^ {(2m- 1) \ pi {\ rm {i}} \ tau -2 \ pi {\ rm {i}} z} \ right].}
Euler's pentagonal theorem results with and :
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{\ displaystyle x = q ^ {\ frac {3} {2}}}
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{\ displaystyle y ^ {2} = - {\ sqrt {q}}}
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{\ displaystyle \ prod _ {m = 1} ^ {\ infty} \ left (1-q ^ {m} \ right) = \ sum _ {n = - \ infty} ^ {\ infty} (- 1) ^ {n} q ^ {(3n ^ {2} -n) / 2}. \,}
Particularly compact, the triple product leaves with the Ramanujan theta function express
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{\ displaystyle f (a, b) = \ sum _ {n = - \ infty} ^ {\ infty} a ^ {n (n + 1) / 2} \; b ^ {n (n-1) / 2 }}
with . Then there is the triple product identity
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{\ displaystyle | from | <1}
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{\ displaystyle f (a, b) = (- a; ab) _ {\ infty} \; (- b; ab) _ {\ infty} \; (ab; ab) _ {\ infty} = \ prod _ {m = 0} ^ {\ infty} \ left (1 + a (ab) ^ {m} \ right) \, \ left (1 + b (ab) ^ {m} \ right) \, \ left (1 - (ab) ^ {m + 1} \ right)}
with the q- Pochhammer symbol . It was and set.
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{\ displaystyle (a; q) _ {n}}
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{\ displaystyle x = (from) ^ {\ frac {1} {2}}}
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{\ displaystyle y ^ {2} = \ left ({\ tfrac {a} {b}} \ right) ^ {\ frac {1} {2}}}
Much evidence of triple product identity is known. Among other things, EM Wright gave a combinatorial proof.
Another formulation that simply follows from the above is:
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{\ displaystyle \ sum _ {n = - \ infty} ^ {\ infty} q ^ {n ^ {2}} z ^ {n} = \ prod _ {n \ geq 0} \ left (1-q ^ { 2n + 2} \ right) \ left (1 + zq ^ {2n + 1} \ right) \ left (1 + {\ frac {1} {z}} q ^ {2n + 1} \ right)}
literature
George E. Andrews : A simple proof of Jacobi's triple product identity. In: Proceedings of the American Mathematical Society . Volume 16, 1965, pp. 333-334, doi: 10.1090 / S0002-9939-1965-0171725-X .
Tom M. Apostol : Introduction to Analytic Number Theory. Springer, New York NY et al. 1976, ISBN 0-387-90163-9 , p. 319.
Godfrey H. Hardy , Edward M. Wright : An Introduction to the Theory of Numbers. 4th edition (reprint). Clarendon Press, Oxford 1975, ISBN 0-19-853310-7 , p. 228 ff.
Edward M. Wright: An enumerative proof of an identity of Jacobi. In: Journal of the London Mathematical Society . Volume 40, 1965, pp. 55-57, doi: 10.1112 / jlms / s1-40.1.55 .
Web links
Individual evidence
^ Herbert Wilf: The number theoretical content of the Jacobi triple product identity. (pdf)
↑ In this form also in GHH Hardy, EM Wright: An Introduction to Theory of Numbers. 4th edition. 1975, p. 282.
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