v-theta Function Identities and Discrete Fourier Transform

R. A. Malekar

National Defence Academy, Khadakwasala,Pune-411023, INDIA.
rajanish1973@gmail.com
(Received on: October 15, Accepted: October 24, 2017)

ABSTRACT

The extensions of classical identities of Jacobi theta functions are obtained corresponding to the v-theta functions. These identities are obtained using the properties of eigenvectors corresponding to the discrete Fourier transform in terms of linear combinations of v-theta functions. In particular these are classical Jacobi theta function identities for v = 1 corresponding to the Φ(2).

MSc 2010 Classification: 33XX, 65T50, 15A18.
Keywords:discrete Fourier transform, v-theta function, Rieman's v-theta function identity, Watson's v-theta function identity.

INTRODUCTION

There are various generalizations of classical Jacobi theta functions in literature starting with the works of Ramanujan and others1,8. Matveev12 has defined v-theta functions which are periodic and extensions of classical Jacobi theta functions. The classical Jacobi theta functions are the particular case of these v-theta functions for v = 1. The eigenvectors of the discrete Fourier transform (DFT) are expressed as a linear combinations in terms of v- theta functions12. This manuscript discusses the identities between v-theta functions which gives extensions of the identities between classical Jacobi theta functions. The v-theta functions are not doubly periodic, hence the techniques we use to derive the identities are similar to the one used in9,10,11. This technique is different than the methods used in deriving the classical identities which are based on the zeros of Jacobi theta functions or infinite product representation3. The basic notations adopted in this paper and some preliminary results are presented in the next section. The identities of theta functions corresponding to the DFT Φ (2) are discussed in the Section 3.

PRELIMINARY RESULTS

The matrix Φ(n) corresponding to the DFT of size n is given by

It is clear from the definition that Φ4 = I. The multiplicities corresponding to the given values of the DFT Φ (n) are well known and are given by (see12).

where mk is multiplicity of the ik . Let τ be complex number with positive imaginary part then generalized v-theta function is defined by

θ(x, τ , u) is an entire function of x satisfying the relation θ(x+1, τ , μ) = θ(x, τ , μ) and it is periodic function of x . This function satisfies the partial differential equation

The u-theta function reduces to the usual theta function for u = 1 . The u-theta function with character a,b is given by &thaeta;a,b(x, τ , u)

The zeros of u-theta function is same as the Classical Jacobi theta functions. The Classical Jacobi theta functions are periodic with period 1,τ . The notations in the manuscript are different than the q-series notation in the literature3. The classical Jacobi theta functions may be represented by

Matveev12 has proved the following important theorem which will be used in the following sections.

The proof of the above theorem follows from the fact that ∅4 = I 12 . The above formula is used to derive the extensions of some of the well known classical identities of the Jacobi theta functions to the generalized u-theta functions. These identities are explored for the DFT ∅ (2) in this manuscript. The extensions of well known fourth order identity and Watson addition formula are obtained.

u-THETA FUNCTIONS AND THE DFT ∅ (2)

This section some identities of fourth order are derived between u-theta functions. These identities are natural extensions of classical well known identities between Jacobi theta functions.

Proof: The DFT ∅ (2) has only two eigenvalues +1 and -1. The eigenvector corresponding to eigenvalue +1 is given by

The eigenvector corresponding to eigenvalue -1 is given by

This gives the following two identities

The identities (10), (11) are equivalent to

The identities (12), (13) gives

Consider the eigenvectors v(x, τ, θ, i), v(x + 1, τ, θ, u) corresponding to the eigen value +1. The 2 x 2 minor formed by the determinant of these eigenvectors is zero.

Using equations (10), (11) the terms with coefficients of 2√2 cancel each other out. This gives

The identities (14) and (15) leads to the required fourth order identity between u-theta functions.

Proof : At u = 1 in (7) and replace x by 2x and τ by 4 τ , we get

At the null values we have

Let m + n = n1 and m-n = n2, so that n1 and n2 are of the same parity. Rewriting the above equation in terms of n1 and n2 leads to the following formulae

These are well known Landen type transformations. Using the similar argument, we have

By (20), (21) in (18), the fourth order identity of theta constants follows. In the next theorem we derive Watson addition formula for u-theta functions. The classical Watson addition formula follows as a particular case.

Proof : Using (10) , (11)

From (8) and (9), we have

are eigenvectors of ∅ (2) corresponding to eigenvalues +1 with multiplicity one . Therefore

This gives

In this expression consider the terms with 2√2 as coefficient. Using formulas (23) and (24) we have

It is clear that A+B-C-D = ∅. Hence all the terms with coefficients 2√2 cancel each other out. Equation (25) becomes

Proof: At u = 1 in (22) the identity reduces the identity between Jacobi theta function. The complete details of the proof can be seen in9.
The next theorem we give Riemann's identity for At u-theta functions. The u -theta functions satisfies the following identity. All theta functions involved are u-theta functions.

Proof : The u-theta identities from (10), (11) we have

These set of equations gives

Changing the variables (x, y) to (u, v) in (32)

Using (32), (33), we get the required formula.
In the next result we show the Riemann's identity between Jacobi theta functions as the particular case of (27).

Proof : At u = 1 in (27) the identity reduces to the identity between Jacobi theta functions. The complete details of the proof are given in10.

For simplicity if we do a change of the variable as follows

Then the particular restrictions on parameters n; m; p; q of the summation above exactly means that the resulting n1, m1, p1, q1 are integers. Also observe that we have the identities :

The beautiful account of identities derived from Riemann identity (35) are given in2.
This includes all the fourth order identities of theta functions. This shows that all these identities have an extensions in the form of u- theta functions. The above method shows that they can be derived using the techniques discussed in the manuscript.

CONCLUSION

This manuscript has discussed the identities of u-theta functions corresponding to the DFT ∅(2). The classical identities of the Jacobi theta functions corresponds to the u = 1. It would be interesting to study these identities corresponding to u = 2. There is a natural extensions of these identities for the DFT ∅(3) as discussed in9,11.

REFERENCES

  1. B C Berndt, S Bhargava, F Garvan, Ramanujan theories of elliptic functions to alternative bases, Trans American Mathematical Soc, Vol 347, (11) 4163-4244 (1995).
  2. D Mumford, Tata lectures on theta I, Birkhauser (1983).
  3. E T Whittaker, G.N. Watson, A course of Modern Analysis, fourth ed. Cambridge University Press (1927).
  4. H Mckean, V Moll, Elliptic Curves, Cambridge University Press(1999).
  5. D Galetti, M A Marchiolli, Discrete Coherent states and probability distributions in finite dimensional spaces, Annals of Physics, 249, 454-480 (1996).
  6. M L Mehta, Eigen values and eigenvectors of finite Fourier transform, J Math. Physics 28 (4), 781-785 (1987).
  7. M Ruzzi, Jacobi theta functions and discrete Fourier transform, J Math. Physics (47), 1-10 (2006).
  8. P Choon Toh, Generalized m-th Order Jacobi Theta Functions and the Macdonald identities, Int. J. Number Theory, 4, 461-474 (2008).
  9. R A Malekar, H Bhate , Discrete Fourier transform and Jacobi theta function identities. J Math. Phy 51, 023511(2010).
  10. R A Malekar, H Bhate, Discrete Fourier transform and Riemann identities for theta functions. Applied Mathematics Letters 25 (2012).
  11. R A Malekar, Discrete Fourier transform ∅ (3) and θa,b (x, τ) a,b ∈1/3Z Journal of Indian Mathematical Soc, Vol 83 Nos (1-2), (2016).
  12. V B Matveev, Intertwining relations between Fourier transform and discrete Fourier transform, the related functional identities and beyond, Inverse Problems 17, 633-657 (2001).
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