An Eisenstein series with half-period ratio and index is defined by
(1)
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where the sum excludes , , and is an integer (Apostol 1997, p. 12).
The Eisenstein series satisfies the remarkable property
(2)
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if the matrix is in the special linear group (Serre 1973, pp. 79 and 83). Therefore, is a modular form of weight (Serre 1973, p. 83).
Furthermore, each Eisenstein series is expressible as a polynomial of the elliptic invariants and of the Weierstrass elliptic function with positive rational coefficients (Apostol 1997).
The Eisenstein series satisfy
(3)
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where is the Riemann zeta function and is the divisor function (Apostol 1997, pp. 24 and 69). Writing the nome as
(4)
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where is a complete elliptic integral of the first kind, , is the elliptic modulus, and defining
(5)
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we have
(6)
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(7)
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where
(8)
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(9)
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(10)
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where is a Bernoulli number. For , 2, ..., the first few values of are , 240, , 480, -264, , ... (OEIS A006863 and A001067).
The first few values of are therefore
(11)
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(12)
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(13)
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(14)
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(15)
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(16)
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(17)
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(Apostol 1997, p. 139). Ramanujan used the notations , , and , and these functions satisfy the system of differential equations
(18)
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(19)
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(20)
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(Nesterenko 1999), where is the differential operator.
can also be expressed in terms of complete elliptic integrals of the first kind as
(21)
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(22)
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(Ramanujan 1913-1914), where is the elliptic modulus. Ramanujan used the notation and to refer to and , respectively.
Pretty formulas are given by
(23)
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(24)
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where is a Jacobi theta function.
The following table gives the first few Eisenstein series for even .
The notation is sometimes used to refer to the closely related function
(25)
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(26)
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(27)
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(28)
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(29)
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(OEIS A103640), where is a Jacobi elliptic function and
(30)
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is the odd divisor function (Ramanujan 2000, p. 32).