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Lambert series and Liouville's identities

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Alaca A., Alaca Ş., McAfee E., Williams K.

Instytut Matematyczny Polskiej Akademii Nauk

The relationship between Liouville's arithmetic identities and products of Lambert series is investigated. For example it is shown that Liouville's arithmetic formula for the sum $∑_{{(a,b,x,y) ∈ ℕ ⁴ \atop ax+by=n}} (F(a-b) - F(a+b))$, where n ∈ ℕ and F: ℤ → ℂ is an even function, is equivalent to the Lambert series for $(∑_{n=1}^{∞} (qⁿ/(1-qⁿ))sin nθ)²$ (θ ∈ ℝ, |q| < 1) given by Ramanujan.

Ograniczanie wyników

1 Alaca A.

1 Alaca Ş.

1 McAfee E.

1 Williams K.

1 2007

Wyniki wyszukiwania

Lambert series and Liouville's identities