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**Tewlede G/Egziabher ^{1}, Hunduma Legesse Geleta^{2*}, Abdul Hassen^{3}**

^{1}Department of Mathematics, Ambo University, Ambo, Ethiopia

^{2}Department of Mathematics, Addis Ababa University, Addis Ababa, Ethiopia

^{3}Department of Mathematics, Rowan University, New Jersey, USA

- *Corresponding Author:
- Hunduma Legesse Geleta

Department of Mathematics, Addis Ababa University, Addis Ababa, Ethiopia

**E-mail:**tewelde.geber@ambou.edu.et

**Received:** 21-Feb-2024, Manuscript No. JSMS-24-128011; **Editor assigned:** 23-Feb-2024, Pre QC No. JSMS-24-128011 (PQ); **Reviewed:** 08-Mar-2024, QC No. JSMS-24-128011; **Revised:** 04-Sep-2024, Manuscript No. JSMS-24-128011 (R); **Published:** 11-Sep-2024, DOI: 10.4172/JSMS.10.3.001

**Citation:** Geleta HL, et al. Automorhpic Integrals with Rational Period Functions and Arithmetical Identities. RRJ Stats Math Sci. 2024;10:001.

**Copyright:** © 2024 Geleta HL, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution and reproduction in any medium, provided the original author and source are credited.

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In 1961, Chandrasekharan and Narasimhan showed that for a large class of Dirichlet series the functional equation and two types of arithmetical identities are equivalent. In 1992, Hawkins and Knopp proved a Hecke correspondence theorem for modular integrals with rational period function on theta group. Analogous to Chandrasekharan and Narasimhan, in 2015 Sister Ann M. Heath has shown that the functional equation in Hawkins and Knopp context and two type of arithmetical identities are equivalent. She considered the functional equation and showed its equivalence to two arithmetical identities associated with entire modular cusp integrals involving rational period functions for the full modular group. In this paper we extend the results of Sister Ann M. Heath to entire automorphic integrals involving rational period functions on discrete Hecke group.

Recurrence; Bernoulli numbers; Bernoulli polynomials; Hypergeometric Bernoulli numbers; Polynomials

Let {λ_{n}} and {μ_{n}} be two strictly increasing sequences of positive real numbers diverging to ∞, as n → ∞ and let {a_{n}} and
{b_{n}} be two sequences of complex numbers not identically zero. Consider the Dirichlet series φ and ψ defined by

φ(s) = $\frac{\mathrm{an}}{\mathrm{\lambda sn}}$ and ψ(s) = $\frac{\mathrm{bn}}{\mathrm{\mu sn}}$

with finite abscissas of absolute convergence σ_{a} and σ_{b}, respectively. Suppose that φ and ψ satisfy the functional equation

Γ(s)φ(s) = Γ(δ − s)ψ(δ − s), (1.1)

where δ>0. Chandrasekharan and Narasimhan [1] showed that the functional equation (1.1) is equivalent to the following arithmetical identities (1.2) and (1.3).

$\frac{1}{\mathrm{\Gamma (\rho \; +\; 1)}}$
a_{n}(x − λ_{n})^{ρ} =
$\left(\frac{1}{\mathrm{2\pi}}\right)\rho $
b_{n}
$\left(\frac{\mathrm{x}}{\mathrm{\mu n}}\right)$ \frac{\mathrm{\delta +\rho}}{2}$J$_{δ+ρ}{4π√μ_{n}x} + Q_{ρ}(x), (1.2)

where x > 0, ρ ≥ 2β − δ − ½ , J_{ν} (z) denotes the usual Bessel function of the first kind of order ν,

Q_{ρ}(x) =
$\frac{1}{\mathrm{2\pi i}}$
∮_{c}
$\frac{\mathrm{\Phi (s)(2\pi )sxs+\rho}}{\mathrm{\Gamma (s\; +\; \rho \; +\; 1)}}ds,$| b_{n} | μ^{-β}_{n} < ∞.
$\left(-\frac{1}{\mathrm{s}}\frac{\mathrm{d}}{\mathrm{ds}}\right)\rho $
$[\frac{1}{\mathrm{s}}$a_{n}e^{-s}√λ_{n}
^{3δ+ρ}Γ(δ+ρ+½)π^{δ-½}
$\frac{\mathrm{bn}}{\mathrm{(s\xb2+16\pi \xb2\mu n)\delta +\rho +\xbd}}+\; R\rho (s),\; (1.3)$

where Re s > 0, ρ is non-negative integer satisfying ρ ≥ β − δ − ½ and

R_{ρ}(s) =
$\frac{1}{\mathrm{2\pi i}}$
∮_{c}
$\frac{\mathrm{\Phi (z)(2\pi )z\Gamma (2z=2\rho +1)2-\rho}}{\mathrm{\Gamma (z\; +\; \rho \; +\; 1)}}s-2z-2\rho -1dz$
.

Note that if β>0, then identity (1.3) holds for ρ satisfying ρ ≥ β − δ − ½ ρ ∈ Z ≥ 0.

In [3], Hawkins and Knopp proved a Hecke correspondence theorem for modular integrals with rational period functions
on Γ_{θ}, (generated by Sz = z + 2 and Tz = −1/z), a subgroup of the full modular group Γ(1). In their work, the functional
equation takes the form

Φ(2k − s) − i^{2k}Φ(s) = R_{k}(s), (1.4)

where Φ(s) =
$\left(\frac{\mathrm{2\pi}}{\mathrm{\lambda n}}\right)-s\Gamma (s)$a_{n}n^{-s}

is associated with a modular relation involving rational period function q(z) of the form

F(z + λ) = F(z) and z^{−2k} F
$\left(\frac{\mathrm{-1}}{\mathrm{z}}\right)$
= F(z) + q(z), (1.5)

where λ = λ_{n} = 2 cos
$\left(\frac{\mathrm{\pi}}{\mathrm{n}}\right)$
, with 3 ≤ n ∈ ℕ ∪ {∞} and 2k ∈ ℤ.

Analogous to Chandrasekharan and Narasimhan, Sister Ann M. Heath [4] showed that the functional equation in the Hawkins and Knopp context (1.4) and the arithmetical identities are equivalent. To prove her results, she used the fact that the correspondence theorem between the functional equational (1.4) and the associated entire integrals form with rational period function for the full modular group Γ(1). We [9] also used techniques of Chandrasekharan and Narasimhan to prove results analogous to those of Sister Ann M. Heath and established equivalence of two arithmetical identities with functional equation associated with automorphic integrals involving log-polynomial-period functions on the discrete Hecke group.

In this paper we use the techniques of Chandrasekharan and Narasimhan [1] and extend the results of Sister Ann M. Heath [4] to entire automorphic integrals involving rational period functions on discrete Hecke group G(λ). This paper is organized as follows: In section two we review some results concerning, Heck groups, automorphic integrals with rational period function and present some preliminary results. In section three we present our main results with their proofs.

**Preliminaries**

In this section we review some terms and results that are useful in the coming sections.

Recall that the Hecke group G(λ), where λ ∈ R^{+}, is defined as the subgroup of SL_{2}(R) given by

Equivalently G(λ) is generated by the linear fractional transformations S(z) = z + λ and T(z) = − $\frac{1}{\mathrm{z}}$ . The element of G(λ) act on the Riemann sphere as linear fractional transformation, that is Mz = $\frac{\mathrm{az+b}}{\mathrm{cz+d}}$ for

M = $\left(\genfrac{}{}{0ex}{}{\mathrm{a\; b}}{\mathrm{c\; d}}\right)$ ∈ G(λ), and z ∈ ℂ ∪{∞},

thus M and −M can be identified with the same linear fractional transformations. Hecke [6] showed the group G(λ) is discrete (operates discontinuously) as a set of linear fractional transformations on the upper half plane

ℋ = {z = x + iy : y > 0} if and only if either λ>2 or λ = λ_{p}:= 2cos
$\left(\frac{\mathrm{\pi}}{\mathrm{n}}\right)$
, with 3 ≤ p ∈ ℕ ∪{∞}. For λ ≥ 2 and λ=λ_{p}, we have the following relations respectively,

T^{2} = −I, T^{2} = (S_{λp} T)^{p} = −I.

It is clear that G(λ_{3}) = G(1) = Γ(1) is the full modular group, and G(λ_{∞}) = Γ_{θ} is the familiar theta group.

Suppose F (z) is a meromorphic function in the upper half planHe t hat satisfies (1.5). Further assume that F has Fourier series expansion of the form

F(z) = a_{m}e^{2πimz/λ}, (2.1)

where ℑz = y > y_{o} ≥ 0 and ν ∈ ℤ. The function F is called an automorphic integral of weight 2k for the Hecke group
G(λ), with rational period function (RP F ) , q(z). If q ≡ 0 then F is an automorphic form of weight 2k on G(λ).
If F is an automorphic integral and holomorphic in H (that is, ν ≥ 0) and satisfies the growth condition

| F(z) | ≤ C (| z |^{α} + y^{−β}) , ℑ(z) = y>0,

for some constants C, α, β, > 0, and z ∈ ℋ one can show that the cofficients a_{m} in (2.1) satisfy

a_{m} = 𝒪(m^{β}), m → ∞.

In this case, F is called an entire automorphic integral of weight 2k on G(λ) with rational period function q.

For M = $\left(\genfrac{}{}{0ex}{}{\mathrm{*\; *}}{\mathrm{c\; d}}\right)$ ∈ G(λ) the stroke or slash operatoris defined by

F | M := F |^{M}_{2k} = (cz + d)^{−2k} F(Mz).

Thus the second automorphic relation (1.5) can be expressed as F |T = F + q. In general, for any M in G(λ) there is a
corresponding period function qM such that F |M = F + q_{M} . The slash operator satisfies F |M_{1}M_{2} = (F |M_{1})|M_{2} for
M_{1}, M_{2} ∈ G(λ) and hence the family of periodic functions{ q_{M} : M ∈ G(λ} are related by

q_{M1M2} = q_{M1}| M2 + q_{M2} , M1, M2 ∈ G(λ). (2.2)

Using the relation T² = −I,(2.2 imposes a relation on the (RPF) q,

q|T + q = 0. (2.3)

And using the relation (S_{λp}T)^{p} = −I for tλ = λ_{p} = 2 cos
$\left(\frac{\mathrm{\pi}}{\mathrm{p}}\right)$
, p ∈ ℤ, p ≥ 3, imposes another condition on (RPF) q,

q |
(S_{λp}T)^{p−1} + q |
(S_{λp}T)^{p−2} + · · · + q |
(S_{λp}T)
+ q = 0. (2.4)

Marvin Knopp [7] proved that the finite poles of a rational period function on Γ(1) are only at 0 or real quadratic
irrationals. He also showed thaqt iisf a RPF of weightk 2k > 0 with poles in Q, then for some constanαts α_{0}, α_{1} ∈ C,

q(z) = $\{\genfrac{}{}{0ex}{}{\mathrm{\alpha 0$ \left(1\; -\frac{1}{\mathrm{z2k}}\right)$if\; k\; 1,}}{\mathrm{\alpha 0$ \left(1\; -\frac{1}{\mathrm{z2}}\right)+\frac{\mathrm{\alpha 1}}{\mathrm{z}}$if\; k\; =\; 1.}}$

Observe that if F(z) ≡ −α_{0}, then (F|T ) (z) = F(z) + q(z) implies that z^{−2k}F
$\left(\frac{\mathrm{-1}}{\mathrm{z}}\right)$
− F(z) = q(z) and hence q(z) = α_{0} (1 − z^{−2k})

Thus we consider q(z) = α_{0} (1 − z^{−2k}) as the trivial period function of weight 2k ∈ R.

The following lemma is stated in the work of Hawkins and Knopp [3], where their underlying groiusp Γ_{θ} and generalized
to the general Hecke group and multiplier system by Hassen [2].

Lemma 2.1. Nontrivial rational period function on the Hecke groups satisfying (2.3) and (2.4) exists only if the weight 2k is an integer.

Wendell-Culp-Ressler in ([8], Lemma 3) showed that the poles of any rational periodic functionq of weight 2k, k ∈ Z^{+} on
With appropriate modifications to fit for the current context of functions on G(λ), the work of Hawkins and Knopp [3] can
be used to state a special form of RP F for the solution of (2.3). This form is given by the following lemma.

Lemma 2.2. For r ∈ ℤ, α_{j} ∈ ℝ \ {0}, C_{r}, C_{rj}∈ ℂ for j = 1, 2 ......., p, let f_{r}(z, 0) = z^{−r} − (−1)^{r }z^{−2k+r}, and

f_{r}(z, α_{j}) = (z − α_{j})^{−r} − (−1)^{r} α_{j}^{−r} z^{−2k+r}
$\left(z\; +\frac{1}{\mathrm{\alpha j}}\right)$
^{−r}. Then

q(z) = C_{r}f_{r}(z, 0) + C_{rj}f_{r}(z, αj) and satisfies q | T + q = 0. (2.5)

Theorem 2.1. Suppose F is an entire automorphic integral function of weight 2k, k ∈ Z^{+} for G(λ) with (RPF ) q(z), where q
has the form described by Lemma 2.2. Suppose further that F has a Fourier series expansion of the form

F(z) = a_{m}e^{2πimz/λ}, with a_{m} = 𝒪(m^{β}) β > 0, m → ∞. (2.6)

φ(s) = a_{m}m^{−s} and Φ(s) =
$\left(\frac{\mathrm{2\pi}}{\mathrm{\lambda}}\right)-s$
Γ(s)φ(s), s = σ + it. (2.7)

Then Φ(s) has a meromorphic continuation to the whole complex plane and can be expressed in the form of Φ(s) = D(s) + D^{0}(s) + E^{0}(s) + E^{H}(s) + E^{B}(s),

where

Moreover, Φ(s) satisfies the functional equation

Φ(2k − s) − i^{2k}Φ(s) = R(s) , where (2.13)

R(s) = E^{B}(2k − s) − i^{2k}E^{B}(s)

and _{2}F_{1} [a, b, c; z] is the hypergeometric function and B(a, b) is the Beta function.

The proof of Theorem 2.1 is similar to that of Hawkins and Knopp [3], with appropriate modifications for the more general group

λ_{p} = 2 cos > 0.
$\left(\frac{\mathrm{\pi}}{\mathrm{p}}\right)$

Corollary 2.1. Suppose Φ(s), D^{0}(s), D(s), E^{0}(s), E^{H}(s) and E^{B}(s) are given as in Theorem 2.1. Then (a) Φ(s) is
bounded uniformly in σ in lacunary vertical strips of the form

S = {s = σ + it : 2k − δ ≤ σ ≤ δ; |t| ≥ to > 0}.

(b) δ in (a) can be chosen so that the poles of Φ(s) lying with in the lines s = (2k − δ) + it and s = δ + it are listed below in the sets;

S_{0} = {0, 2k}, S_{E0} = {2k − L, 2k − L + 1, ..., k − 1, k, k + 1, ..., 2k, ..., L},

S_{H} = {[2k − δ], ..., 0}, S_{B} = {[2k − δ], ..., 2k − L, ..., 2k − 1}.

The poles of Φ(s) in each set arise from D^{0}(s), E^{0}(s), E^{H}(s) and E^{B}(s) respectively. (c) The residues of Φ(s) are given by the formula

Before we state and prove our main results, we state Perron’s formula as Lemma 2.3 below (see [1] for details.) We shall also use the convention of writing

Lemma 2.3. Let σ_{0} be the abscissa of absolute convergence for φ(s) = a_{m}λ^{−s}_{m}and {λm} be a sequence
of positive real numbers tending to ∞ as m → ∞. Then for k ≥ 0, σ > 0 and σ_{0},

where the prime′ on the summation sign indicates that if k = 0 and x = λ_{m} for some positive integer m, then we count
only ½a_{m}.

The evaluation of the integral in (2.18) of Lemma 2.3, we consider a positively oriented rectangular contour formed by

[(2k − σ) − iT, σ − iT ], [σ − iT, σ + iT ], [σ + iT, (2k − σ) + iT ], [(2k − σ) + iT, (2k − σ) − iT ],

use Stirling’s approximation formula for the gamma function, Phragmen-Lindelo¨f theorem [5] and apply Cauchy Residue theorem where in all cases the parameters are choosen appropriately to satisfy the conditions. Note that we use this in several places and cite it as [5].

In this section, we shall use the techniques of Chandrasekharan and Narasimhan in [1] to extend the first result in [4] to entire automorphic integrals on discrete Hecke groups G(λ).

Theorem 3.1. (F irstEquivalence) . Let Φ(s) and R(s) be as in Theorem 2.1. Then the functional equation

Φ(2k − s) − i^{2k}Φ(s) = R(s) (3.1)

is equivalent to the identity

x > 0, ρ ≥ 2β − 2k − ½ , and β is a number for which $\frac{\mathrm{|am|}}{\mathrm{m\beta}}$ < ∞.

Proof. We use [5] to arrive at

Hence by using again [5] (3.3) can now be written as

We now show that the functions A1(x) and A2(x) can be expressed respectively as

Observe that using the functional equation Φ(2k − s) − i^{2k}Φ(s) = R(s), we have

Since Φ(s) = $\left(\frac{\mathrm{2\pi}}{\mathrm{\lambda}}\right)-s$ Γ(s)φ(s), and denoting the first integral by I(x) we see that

where we have used change of variable froms to 2k − s, in the first integral. Letting $\frac{\mathrm{w}}{2}$ = s, ν = 2k + ρ, and simplifying expressions, we obtain

provided that ρ > σ − 2k and σ > 2k, k ∈ Z. Note also that we have used the definition of the Bessel J-function here. Now we evaluate the second integral denoting by H(x) and using the expression for R(x) as follows

Using the properties of the beta function and after some algebraic manipulations, we have

Using the properties of hypergeometric series and simplifying, we get

Since A_{1}(x) = I(x) − H(x), we have proved (3.7) and proceed to computing A_{2}(x) as follows.

From theorem 2.1, Φ(s) has been expressed in terms of D^{0}(s), E^{0}(s), E^{H}(s), and E^{B}(s). Thus A_{2} can be expressed as
follows

Using formula (2.14), (2.15), (2.16)and (2.17) we obtain the following and hence (3.8) is proved.

Finally, we now express the integral in the right-hand side of (3.4) as

and using the expressions for A_{1}(x) and A_{2}(x). [See (3.7) and (3.8) respectively] we obtain

Therefore, for ρ ≥ 0, and $\frac{\mathrm{\rho \; +\; 2k}}{2}$ > δ we get the identity (3.2).

Now we prove the converse of the theoremo. Tthis end supposeF (z) is an entire automorphic integral with ao uFrier series expansion for z ∈ H and satisfies the relatio

F(z) = a_{m}e^{2πimz/λ},

z^{−2k}F
$\left(\frac{\mathrm{-1}}{\mathrm{z}}\right)$
= F(z) + q(z), (3.10)

where q(z) is the rational period function given by Lemma (2.2). Then by (3.10) and the Fourier expansion of F we have

z^{−2k}a_{m}e^{−2πim/λz} = a_{m}e^{2πimz/λ} + q(z).

Letting z = $\frac{\mathrm{iy\lambda}}{\mathrm{2\pi}}$ , y > 0, then we get

To prove the converse it suffices to show that (3.2) implies (3.11). To this end we consider six integrals defining L_{1}(y) · · · , L_{6}(y), corresponding to the six expressions occurring in (3.9). We evaluate all the integrals by interchanging integration
and summation which can be justified

Similarly L_{2}(y) for
$\frac{\mathrm{\rho \; +\; 2k}}{2}$
≤ β and applying formula (??) we get

Thus with simple algebraic manipulations we obtain

Using integration by substitution and the standard integral representation of Γ(s) we get

After evaluating and simplifying we obtain

Using the series representation of the hypergeometric function for λy > 2πα_{j} , we obtain

This series converges absolutely for λy > 2πα_{j}. In a Similar wa,y we compute L_{6} as follows

This series converges absolutely for y >
$\frac{\mathrm{2\pi}}{\mathrm{\lambda \alpha j}}$
. Combining the results of the integrals for L_{1}(y), L_{2}(y), · · · , L_{6}(y)
respectively we have

Thus after simple algebraic manipulations, we see that the identity in (3.9) implies

Recall that the rational periodic function in Lemma 2.1.

Since F(z) has a Fourier expansion of the form a_{m}e^{2πimz/λ} and q
$\left(\frac{\mathrm{iy\lambda}}{\mathrm{2\pi}}\right)$
is represented (3.13), (3.12) may
be written as

Hence by the identity theorem the automorphic transformation

follows for z ∈ H. This concludes the proof of the equivalence of the functional equation to the identity (3.9). Theorem 3.2. (Second Equivalence) Let Φ(s) and R(s) as in Theorem 2.1 then the functional equation

Φ(2k − s) − i^{2k}Φ(s) = R(s) (3.1)

is equivalent to the identity

Proof. By theorem .31 the identity (3.2) is equivalent to the functional equation (.13). Hence to prove this theorem it suffices to show that (3.1) implies (3.2) and (3.2) in turn implies (3.1).

As in Chandrasekharan and Narasimhan ([1] page 9), we multiply (3.3) by e^{−y√x}x^{−½} and integrating with respect to
the variable x from x = 0 to ∞; and further assuming δ > 2k, where Res = δ is the vertical path of integration. Now
choose δ = β + P, where P ∈ Z, and P is large enough to guarantee δ > 2k and δ ∉ Z to have

with ρ+2k ≥ δ + ½ , and y ∈ ℝ^{+}. Chandrasekharan and Narasimhanin [1] for ρ+2k + ½ ≥ β and λ_{n} sequence of
positive real numbers; λ_{n} → ∞, showed the identity

Then put λn = m in (3.5) to have

Since Φ(s) = Γ(s)φ(s) $\left(\frac{\mathrm{2\pi}}{\mathrm{\lambda}}\right)-s$ , where Res > β and $\frac{\mathrm{|am|}}{\mathrm{m\beta}}$ < ∞, the right-hand side of (3.3) becomes

for δ ≥ β. Interchanging the order of integration for ρ ≥ 0 we obtain

Using the properties of Γ function we have

Using [5] we evaluate U (y) and have

Denote the first and second integrals as T_{1}(y) and T_{2}(y) respectively so that U(y) = T_{1}(y) + T_{2}(y). Using the identity, Φ(s) = i^{−2k} (Φ(2k − s) − R(s)) , and substituting in to the integrand of T_{1}(y), we can write as

T_{1}(y) = V_{1}(y) − V_{2}(y),

where V_{1}(y) and V_{2}(y) are given as

Using the substitution ϑ = 2k − s and Φ(ϑ) = $\left(\frac{\mathrm{2\pi}}{\mathrm{\lambda}}\right)-\vartheta $ Γ(ϑ)φ(ϑ), where φ(ϑ) = $\frac{\mathrm{am}}{\mathrm{m\beta}}$ replacing ϑ by −ϑ, we have

The series converges fory > 0, provided 2k + ρ − ½ > β. Since δ ≥ β the series converges absolutely for ρ ≥ δ − 2k + ½.

Replacing the beta function by its equivalent Γ function and replacinRg( s) by its equivalent representation we have

Using [5] we evaluate Q1(y) and obtain

It can be easily shown that the integral on the right-hand side tends to zero as N tends to ∞. Thus evaluating the residue of the poles we have

Again using properties of the Γ function and simplifying we obtain

Using series representation of the confluent hypergeometric function we get

Using the the confluent hypergeometric function of the second kind we have

Thus for ρ ≥ 0, ρ ∈ ℤ, ρ + 2k ≥ δ + ½, δ > 2k and y ∈ ℝ^{+} the functional equation in (3.1) implies

Equating the left-hand and right-hand side expression we obtain

where T_{2}(y) and V_{2}(y) are given by (3.6) and (3.7) respectively. We can also get an explicit formula for V_{2}(y)
from (3.6) and (3.8).

Using the expression for T_{2} from (3.6) and the residue of Φ(s) from theorem 2.1 we have

Now substituting (3.11) into (3.10) for T_{2}(y) and by substituting the respective expressions (3.8), (3.9) into V_{2}(y),
we obtain the identity (3.2) and this completes the proof of implication part.

To prove the converse we only show (3.2) implies (3.1). Multiply (3.2) by e^{y√x}, with Rey > 0 and x > 0 and integrate the
expression along vertical path Res = ϑ, where ϑ > 0. The left hand side (3.2) can be

evaluated using the formula ([1], page 9)

For the right-hand side of (3.2) we compute the integral of each term one by one. So put the first

Using Sterling formula and simplifying we obtain

Put the second term

Using the integral in (??) we have

Using properties of the Γ function we obtain

Using [5] we obtain

Therefore, we conclude that

Next put

For fixed r, −r < θ < −r + ½ and −π/2 ≤ arg $\left(\frac{\mathrm{i\alpha j\lambda \xb2}}{\mathrm{8\pi}}\right)$ , using the integral representation of confluent hyper-geometric function of the second kind we get

By interchanging the order of integration which can be justified we obtain

Computing the last integral and simplifying we obtain

Using properties of the Γ function we obtain

Using [5]) we obtain

Thus using this result for I_{2}(x) we find that

Put the third term as I_{3}(x) which is given by

Evaluating each integrals in I_{3}(x) and simplifying, we get

Finally we evaluate the last term in (3.2) which we denote by I_{4}(x) as

Thus combining the results in (3.13), (3.14), (3.15), (3.16), and (3.17) we obtain

This completes the proof of the converse and therefore, the proof of theorem is completed.

- Chandrasekharan K, et al. Hecke’s functional equation and arithmetical identities. Ann Math. 1961;74:1-23.
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- Heath SA, et al. Dirichlet series with functional equations and arithmetical identities. Ramanujan J. 2016;41:115-46.
- Berndt BC, et al. Hecke’s Theory of Modular forms and Dirichlet series. World Sci Res. 2008.
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- Knopp M, et al. Rational period functions of the modular group II. Glasgow Math J. 1981;22:185-197.
- Resseler W, et al. A Hechek Correspondence Theorem for Automorphic Integrals with symmetric Rational period Functions on Hecke Groups. Franklin and Marshal college, Lancaster. 2008.
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