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MODULAR FORMS-page43

MODULAR FORMS-page43 - p n-p n-1-p n-2 p n-5-1 k p n-1 2...

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39 This gives the Euler identity X r Z ( - 1) r q r (3 r +1) / 2 = Y m =1 (1 - q m ) . (4.23) In particular, we get the following Fourier expansion for the Dedekind’s function η ( τ ): η ( τ ) = q 1 24 X r Z ( - 1) r q r (3 r +1) / 2 . The positive integers of the form n + ( k - 2) n ( n - 1) 2 ,n = 1 , 2 ,... are called k -gonal numbers . The number of beads arranged in the form of a regular k -polygon is expressed by k -gonal numbers. In the Euler identity we are dealing with pentagonal numbers. They correspond to the powers of q when r is negative. The Euler identity (4.23) is one of the series of MacDonald’s identities associated to a simple Lie algebra: X r Z a r,k q r = Y m =0 (1 - q m ) k . The Euler identity is the special case corresponding to the algebra s l (2). Exercises 4.1 Let p ( n ) denote the number of partitions of a positive integer n as a sum of positive integers. Using the Euler identity prove that
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Unformatted text preview: p ( n )-p ( n-1)-p ( n-2) + p ( n-5) + ... + (-1) k p ( n-1 2 k (3 k-1))+ (-1) k p ( n-1 2 k (3 k + 1)) + ... = 0 . Using this identity compute the values of p ( n ) for n ≤ 20. 4.2 Prove the Gauss identity : 2 ∞ Y n =0 (1-x 2 n +2 ) „ ∞ Y n =0 (1-x 2 n +1 ) «-1 = ∞ X r =0 x r ( r +1) / 2 . 4.3 Prove the Jacobi identity : ∞ Y n =1 (1-x n ) 3 = ∞ X r =0 (-1) r (2 r + 1) x r ( r +1) / 2 . 4.4 Using (4 . 2) prove the following identity about Gaussian sums : 1 √ q q-1 X r =0 e-πr 2 p/q = 1 √ p p-1 X r =0 e-πr 2 q/p . Here p,q are two coprime natural numbers.[Hint: Consider the assymptotic of the function f ( x ) = Θ(0; ix + p q ) when x goes to zero.]...
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