Tewlede G/Egziabher1, Hunduma Legesse Geleta2*, Abdul Hassen3
1Department of Mathematics, Ambo University, Ambo, Ethiopia
2Department of Mathematics, Addis Ababa University, Addis Ababa, Ethiopia
3Department of Mathematics, Rowan University, New Jersey, USA
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 {λn} and {μn} be two strictly increasing sequences of positive real numbers diverging to ∞, as n → ∞ and let {an} and {bn} be two sequences of complex numbers not identically zero. Consider the Dirichlet series φ and ψ defined by
φ(s) = and ψ(s) =
with finite abscissas of absolute convergence σa and σb, 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).
an(x − λn)ρ = bn
where x > 0, ρ ≥ 2β − δ − ½ , Jν (z) denotes the usual Bessel function of the first kind of order ν,
Qρ(x) =
where Re s > 0, ρ is non-negative integer satisfying ρ ≥ β − δ − ½ and
Rρ(s) =
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 Γθ, (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) − i2kΦ(s) = Rk(s), (1.4)
where Φ(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
where λ = λn = 2 cos
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 SL2(R) given by
Equivalently G(λ) is generated by the linear fractional transformations S(z) = z + λ and T(z) = −
M =
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 = λp:= 2cos
T2 = −I, T2 = (Sλp T)p = −I.
It is clear that G(λ3) = G(1) = Γ(1) is the full modular group, and G(λ∞) = Γθ 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) = ame2πimz/λ, (2.1)
where ℑz = y > yo ≥ 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 am in (2.1) satisfy
am = 𝒪(mβ), m → ∞.
In this case, F is called an entire automorphic integral of weight 2k on G(λ) with rational period function q.
For M =
F | M := F |M2k = (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 + qM . The slash operator satisfies F |M1M2 = (F |M1)|M2 for M1, M2 ∈ G(λ) and hence the family of periodic functions{ qM : M ∈ G(λ} are related by
qM1M2 = qM1| M2 + qM2 , 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 (SλpT)p = −I for tλ = λp = 2 cos
q | (SλpT)p−1 + q | (SλpT)p−2 + · · · + q | (SλpT) + 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 α0, α1 ∈ C,
q(z) =
Observe that if F(z) ≡ −α0, then (F|T ) (z) = F(z) + q(z) implies that z−2kF
Thus we consider q(z) = α0 (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 Γθ 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, αj ∈ ℝ \ {0}, Cr, Crj∈ ℂ for j = 1, 2 ......., p, let fr(z, 0) = z−r − (−1)r z−2k+r, and
fr(z, αj) = (z − αj)−r − (−1)r αj−r z−2k+r
q(z) = Crfr(z, 0) + Crjfr(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) = ame2πimz/λ, with am = 𝒪(mβ) β > 0, m → ∞. (2.6)
φ(s) = amm−s and Φ(s) =
Then Φ(s) has a meromorphic continuation to the whole complex plane and can be expressed in the form of Φ(s) = D(s) + D0(s) + E0(s) + EH(s) + EB(s),
where
Moreover, Φ(s) satisfies the functional equation
Φ(2k − s) − i2kΦ(s) = R(s) , where (2.13)
R(s) = EB(2k − s) − i2kEB(s)
and 2F1 [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
λp = 2 cos > 0.
Corollary 2.1. Suppose Φ(s), D0(s), D(s), E0(s), EH(s) and EB(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;
S0 = {0, 2k}, SE0 = {2k − L, 2k − L + 1, ..., k − 1, k, k + 1, ..., 2k, ..., L},
SH = {[2k − δ], ..., 0}, SB = {[2k − δ], ..., 2k − L, ..., 2k − 1}.
The poles of Φ(s) in each set arise from D0(s), E0(s), EH(s) and EB(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 σ0 be the abscissa of absolute convergence for φ(s) = amλ−smand {λm} be a sequence of positive real numbers tending to ∞ as m → ∞. Then for k ≥ 0, σ > 0 and σ0,
where the prime′ on the summation sign indicates that if k = 0 and x = λm for some positive integer m, then we count only ½am.
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) − i2kΦ(s) = R(s) (3.1)
is equivalent to the identity
x > 0, ρ ≥ 2β − 2k − ½ , and β is a number for which
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) − i2kΦ(s) = R(s), we have
Since Φ(s) =
where we have used change of variable froms to 2k − s, in the first integral. Letting
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 A1(x) = I(x) − H(x), we have proved (3.7) and proceed to computing A2(x) as follows.
From theorem 2.1, Φ(s) has been expressed in terms of D0(s), E0(s), EH(s), and EB(s). Thus A2 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 A1(x) and A2(x). [See (3.7) and (3.8) respectively] we obtain
Therefore, for ρ ≥ 0, and
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) = ame2πimz/λ,
z−2kF
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−2kame−2πim/λz = ame2πimz/λ + q(z).
Letting z =
To prove the converse it suffices to show that (3.2) implies (3.11). To this end we consider six integrals defining L1(y) · · · , L6(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 L2(y) for
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 > 2παj , we obtain
This series converges absolutely for λy > 2παj. In a Similar wa,y we compute L6 as follows
This series converges absolutely for y >
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 ame2πimz/λ and q
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) − i2kΦ(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√xx−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 λn sequence of positive real numbers; λn → ∞, showed the identity
Then put λn = m in (3.5) to have
Since Φ(s) = Γ(s)φ(s)
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 T1(y) and T2(y) respectively so that U(y) = T1(y) + T2(y). Using the identity, Φ(s) = i−2k (Φ(2k − s) − R(s)) , and substituting in to the integrand of T1(y), we can write as
T1(y) = V1(y) − V2(y),
where V1(y) and V2(y) are given as
Using the substitution ϑ = 2k − s and Φ(ϑ) =
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 T2(y) and V2(y) are given by (3.6) and (3.7) respectively. We can also get an explicit formula for V2(y) from (3.6) and (3.8).
Using the expression for T2 from (3.6) and the residue of Φ(s) from theorem 2.1 we have
Now substituting (3.11) into (3.10) for T2(y) and by substituting the respective expressions (3.8), (3.9) into V2(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 ey√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
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 I2(x) we find that
Put the third term as I3(x) which is given by
Evaluating each integrals in I3(x) and simplifying, we get
Finally we evaluate the last term in (3.2) which we denote by I4(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.
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