A Sequence of Modular Forms Associated with Higher-Order Derivatives of Weierstrass-Type Functions

Filomat 30:2 206, 3253 3263 DOI 0.2298/FIL62253A Published by Faculty of Sciences and Mathematics, University of iš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat A Sequence of Modular Forms Associated with Higher-Order Derivatives of Weierstrass-Type Functions A. Ahmet Aygunes a, Yılmaz Simsek a, H. M. Srivastava b a Department of Mathematics, Faculty of Art and Science, University of Akdeniz, TR-07058 Antalya, Turkey b Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3R4, Canada and China Medical University, Taichung 40402, Taiwan, Republic of China Abstract. In this article, we first determine a sequence { f n τ} n of modular forms with weight 2 n k + 42 n n ; k \ {}; := {, 2, 3, }. We then present some applications of this sequence which are related to the Eisenstein series and the cusp forms. We also prove that higher-order derivatives of the Weierstrass type -functions are related to the above-mentioned sequence { f n τ} n of modular forms.. Introduction, Definitions and Preliminaries Throughout this paper, we let := {, 2, 3, } and Z := {0, ±, ±2, }. We also let R and C denote the sets of real and complex numbers, respectively. We shall make use of the following definitions and notations. Let H denotes the right-half complex plane, that is, H := { z : z C and Iz > 0 }. Let z, z 2 C \ {0} with z z 2 R. Then Ω = Ωz, z 2 := {w : w = m z + m 2 z 2 m, m 2 Z} be a lattice in C, z z 2 H and Ω = Ω z, z 2 = Ωz, z 2 \ {0, 0}. 200 Mathematics Subject Classification. Primary B68, M06, 33B5; Secondary 33E99, 65D7 Keywords. Eisenstein series; Modular functions; Modular forms; Cusp forms; Weierstrass -function; Weierstrass type - functions; Laurent series; Mittag-Leffler meromorphic functions. Received: September 204; Accepted: 03 December 204 Communicated by Dragan S. Djordjević Email addresses: aygunes@akdeniz.edu.tr A. Ahmet Aygunes, ysimsek@akdeniz.edu.tr Yılmaz Simsek, harimsri@math.uvic.ca H. M. Srivastava

A. Ahmet Aygunes et al. / Filomat 30:2 206, 3253 3263 3254 Then the Weierstrass -function is defined by see, for example,, and 20 u; w = u + 2 u w.. 2 w 2 w Ω ext, if we define the Eisenstein series G2k, w of weight 2k by the series: G2k, w = k \ {},.2 w Ω w 2k the following relation involving Laurent series holds true between the Eisenstein series G2k, w and the Weierstrass -function u; w: u; w = u + 2k + G2k + 2, wu 2k.3 2 k= 0 < u < min{ w : w 0}. In Section 2, we shall find it to be convenient to recall a family of Weierstrass-type functions which were introduced and investigated by Chang et al. see, for details, 3 4; see also 2. The above-defined Weierstrass -function u; w will then turn out to be a special case of the Weierstrass-type functions. aturally, therefore, our aim will be to find a relationship between the r th derivative of the Weierstrasstype functions and the inhomogeneous Eisenstein series G,k,a for fixed a, which was defined by Schoeneberg. As a matter of fact, Simsek 5 already gave a special relation between the k 2th derivative of the Weierstass -function and the inhomogeneous Eisenstein series G,k,a see also 4. In Section 3, we derive a general formula involving functional sequences which consist of modular forms with weight 2 n+ k + 42 n n ; k \ {}. Finally, in Section 4, we consider an interesting special case of the formula derived in Section 4 which involves functional sequences consisting of modular forms with weight 4k + 4. By using this special case, we obtain Fourier series expansions of modular forms with weights 2 and 6. 2. The Derivatives of the Weierstrass-Type Functions Following the earlier investigations by Chang et al. 3 and 4 see also Wu et al. 2, the Weierstrasstype functions are defined by u; w = u + w Ω u w w n. The function u; w is a Mittag-Leffler meromorphic function in the complex u-plane and has poles of the second order at the points of Ω. Recently, Aygunes and Simsek 2 investigated the behavior of the Weierstrass-type functions under the Hecke operators. We now compute the first-, the second- and the third-order derivatives of u; w as follows: u; w = u + w Ω u w, + + u; w = + u +2 + w Ω u w +2

and + + 2 u; w = u +3 A. Ahmet Aygunes et al. / Filomat 30:2 206, 3253 3263 3255 w Ω + + 2 u w +3, respectively. Here, as usual, the rth derivative of u; w with respect to the argument u is denoted by u; w. By induction, we obtain the following theorem. r Theorem. Let n, r \ {}. Then r + r! u; w = r. 2.! +r w Ω u w Let and k be natural numbers with and k 3. Following the notations of Schoeneberg, suppose also that m, a and z are matrices given by where Also let m = m m 2, a = a a 2 and z = m, m 2, a, a 2 Z and z, z 2 C. m T = m m 2, so that, obviously, w = m T z = m m 2 z z 2 z = m z + m 2 z 2. z 2 Thus, as a function of u, the above definition of the Weierstrass-type function u; w can be rewritten as follows: u; w = u + u m T z, m T z where z z 2 H m,m 2 Z Z and the prime * on the above summation sign indicates that the term corresponding to, m T z = 0 is to be omitted. For a fixed a, the homogeneous Eisenstein-type series G,k,a can be defined by G,k,a z = m a m 0 m T z k ; k \ {, 2}, 2.2 where we have used the notation m a in the following sense: m m 2 a a 2 mod

A. Ahmet Aygunes et al. / Filomat 30:2 206, 3253 3263 3256 or, equivalently, m a mod and m 2 a 2 mod. Schoeneberg, p. 55, Theorem proved that, for natural numbers and k 3, the Eisenstein series G,k,a z defined by 2.2 is an entire modular form of level and weight k. Moreover, the series in 2.2 converges absolutely for all z see, for details,. If, in the equation 2., we replace u by at z with 0 a 0 0 :=, 0 then we find that r a T z ; w r + r! =! Hence we have r a T z ; w m,m 2 Z Z = r +r + r!! m,m 2 Z Z a z +a 2 z 2 m z + m 2 z 2. +r a + m z + a 2 + m 2 z 2 +r. The prime * on each of the above the above summation signs indicates that the term corresponding to is to be omitted. By setting l = a + m and l 2 = a 2 + m 2 a + m z + a 2 + m 2 z 2 = 0 in the above equation, we get r a T z ; w = r +r + r!! w, +r w Ω which readily yields r a T z ; w = r +r + r!! where, for convenience, L := l l 2 L a and L T = l l 2. We are thus led to the following theorem. Theorem 2. Let n, r \ {}. If L T z +r, then u = at z and a 0, r a T z ; w = r +r + r! G,+r,a z + r \ {, 2}. 2.3!

A. Ahmet Aygunes et al. / Filomat 30:2 206, 3253 3263 3257 Remark. If we take n = in the assertion 2.3 of Theorem 2, then we arrive at the following result cf., p. 57, Eq. 6; see also 5: a r T z ; w = r r +! r+2 G,r+2,a z. 3. The Set of Main Theorems By using the derivative of modular forms, it is possible to obtain several useful formulas which can then be applied in order to derive various other modular forms. In recent years, many authors have investigated and studied this subject-area see, for example, 7, 0 and 6. In particular, Koblitz 7 gave a formula which leads to the modular form of weight k + 2 for the following modular group: Γ = SL 2 Z as asserted by Theorem 3 below see 7 and 6. Theorem 3. Let f z be a modular form of weight k for the modular group If Γ = SL 2 Z. hz = 2πi d dz { f z} k 2 E 2z f z, then hz is a modular form of weight k + 2 for the modular group Γ, where E 2 z is an Eisenstein series. Rankin 0 proved the following result. Theorem 4. cf. 0, p. 4, Theorem 3 For an even integer k \ {}, k + 3 G k+2 τ = 2 d k + dτ {G kτ} + 2 k 2 k G 2v τg k+2 2v τ. 2v Remark 2. For even integer k 6, we get cf. and 0 k + 3 k 4 G k+2 τ = 2kk v= k 2 2 k 2 G 2v+2 τg k 2v τ, 3. 2v v= which can indeed be proven by means of the differential equation satisfied by the Weierstrass -function see, for details, and 0. Theorem 5. cf. 0, p. 4, Theorem 3 If the integer k is even and greater than 2, then G k + k τ + 4G 2τG k τ 2kG 2 τg kτ = 3 G k 2 τ + 2 <v< k 2 k 2v G 2v 2 τg k 2v+2τ 0<v< k 2 k G 2v 2v τg k 2v τ.

A. Ahmet Aygunes et al. / Filomat 30:2 206, 3253 3263 3258 Rankin s form of the normalized Eisenstein series can be written as follows see 0: G k τ = B k 2k E kτ k, 3.2 where B k denotes the Bernoulli number of order k see, for details, 9; see also the recent investigations 7 and 8. It is easily seen from 3.2 that E 2k τ = 2 k 2π 2k 2k! 2ζ2k G 2kτ k 3.3 in terms of the Riemann zeta function ζs given by see, for details, 9 n = Rs > s 2 s s and ζs := n= 2 s n= n= n n s Rs > 0; s k+ 2π 2k ζ2k = B 2k k. 3.4 2 2k! It should be remarked in passing that the factor 2 k 2π 2k 2k! is dropped from the right-hand side of 3.3 in the normalization of the Eisenstein series used by many authors including for example Apostol, Eichler and Zagier 6, Koblitz 7 and Silverman 3. In this section, we need the following definitions and theorems which will be used to prove Theorem 6. Our Theorem 6 provides us with a general formula answering a question of Silverman 3, p. 90, Exercise.20 a. Definition. see 3, p. 24 Let k Z and let f τ be a function defined on H. We say that f is weakly modular of weight 2k for Γ if the following two conditions are satisfied: i The function f τ is meromorphic on H; ii The function f τ is such that f Aτ = 2k f τ, where a b A = c d Γ, τ H and Aτ =. Definition 2. see 3, p. 24 A weakly modular function that is meromorphic at is called a modular function. Furthermore, a modular function which is everywhere holomorphic on H including also at is called a modular form. If, in addition, f = 0, then the function f is called a cusp form. Remark 3. According to the work of Silverman 3, p. 24, Remark 3.3, one can see that f τ + = f τ and f = τ 2k f τ. τ

A. Ahmet Aygunes et al. / Filomat 30:2 206, 3253 3263 3259 Therefore, f is a function of q := e 2πiτ. This function f is meromorphic in { q : 0 < q < } and f has the following Fourier expansion: a n q n. n= Remark 4. If the function f is meromorphic at, then it has the following Fourier expansion: n= n 0 a n q n, where n 0 is a positive integer. If the function f is holomorphic at, then it has the following Fourier expansion: a n q n. n=0 In this section, we are interested in addressing the following question which is raised in the book of Silverman 3, p. 90, Exercise.20 a: Let f τ be a modular form of weight 2k. The prove that d { f 2 τ = 2k + f τ } 2 2k f τ d2 dτ dτ { f τ} 3.5 2 is a modular form of weight 4k + 4. Motivated essentially by Silverman s question 3.5, we give our main result in this section, which is stated as Theorem 6 below. For some closely-related earlier works on this subject, the interested reader may be referred to the works by for example Cohen 5 and Rankin 8 and 9. Indeed, by suitably specializing Theorem 6 below, we deduce a number of interesting consequences including 3.2. We also give some examples and corollaries related to Theorem 6. The main theorem of our present investigation can now be stated as follows. Theorem 6. Let n and τ H. Suppose also that f τ be a modular form with weight 2k. Then the sequence fn τ n of modular forms satisfies the following recurrence relation: where f n+ τ = 2 n k + 42 n + f n τ 2 2 n k + 42 n f n τ f n τ, f nτ = d dτ { fn τ } and f n τ = d2 dτ 2 { fn τ }. Proof. Since f n τ is a modular form with weight 2 n k + 42 n n, we have f n = k+42 n f n τ.

A. Ahmet Aygunes et al. / Filomat 30:2 206, 3253 3263 3260 We now compute the derivative of each member of this last equation with respect to τ to prove that f n+ τ is a modular forms with the following weight: 2 n+ k + 42 n. We thus find that { } d f n dτ = a c 2 f n and { } d f n = 2 n k + 42 n c k+42 n f n τ dτ + k+42 n f nτ. ext, since ad bc =, we have f n = 2 n k + 42 n c k+42 n + f n τ On the other hand, we easily see that { } d 2 2c f dτ 2 n = f 3 n We thus obtain { } d 2 f dτ 2 n or, equivalently, + k+42 n +2 f nτ. 3.6 + f 4 n. = 2 n k + 42 n 2 n k + 42 n c 2 k+42 n 2 f n τ 2c f n + 2c 2 n k + 42 n k+42 n f nτ + k+42 n f n τ. + f n = 2 n k + 42 n 2 n k + 42 n c 2 k+42 n +2 f n τ By using 3.6, we have f n + 2c 2 n k + 42 n k+42 n +3 f nτ + k+42 n +4 f n τ. = 2 n k + 42 n 2 n k + 42 n + c 2 k+42 n +2 f n τ + 2c 2 n k + 42 n + k+42 n +3 f nτ + k+42 n +4 f n τ. 3.7

Moreover, by using 3.6 and 3.7, we get 2 n k + 42 n + f n A. Ahmet Aygunes et al. / Filomat 30:2 206, 3253 3263 326 2 = 2 n k + 42 n + { 2 n k + 42 n 2 c 2 + k+82 n +2 f n τ 2 + 2c 2 n k + 42 n + k+82 n +3 f n τ f nτ + + k+82 n +4 f nτ 2 } 3.8 and 2 n k + 42 n f n f n = 2 n k + 42 n { 2 n k + 42 n 2 n k + 42 n + c 2 + k+82 n +2 f n τ 2 + 2c 2 n k + 42 n + + k+82 n +3 f n τ f nτ + + k+82 n +4 f n τ f n τ }. 3.9 Consequently, by applying these last results 3.6 to 3.9, we arrive at the desired assertion given by f n+ = + k+82 n +4 { 2 n k + 42 n + f n τ 2 2 n k + 42 n f n τ f n τ } = + k+42 n f n+ τ. Remark 5. By replacing n by n + in the weight of the modular form f n τ in Theorem 6, we readily see that the modular form f n+ τ has the following weight: 2 n+ k + 42 n. 4. Applications of Theorem 6 Theorem 6 can be shown to have many applications in the theory of the elliptic modular forms. Here we give a few applications related to Theorem 6. Upon setting n = in Theorem 6, we get the following corollary, which was given by Silverman 3, p. 90 and more recently also by Sebbar and Sebbar 2, p. 408. Corollary. Let f τ be a modular form with weight 2k. Then f 2 τ defined by d f 2 τ = 2k + dτ is a modular form with weight 4k + 4. Remark 6. If we put f τ = E 4 τ in 4., we have { } 2 f τ 2k f τ d2 { f τ} 4. dτ2 f 2 τ = 5 d dτ {E 4τ} 2 4E 4 τ d2 dτ 2 {E 4τ}. 4.2

and A. Ahmet Aygunes et al. / Filomat 30:2 206, 3253 3263 3262 ext, by substituting the following well-known derivative formulas for E 4 τ cf. 6, 0 and 2 : d dτ {E 4τ} = 2πi 3 E 2τE 4 τ E 6 τ d 2 dτ {E 4τ} = 5π2 {E2 τ} 2 E 2 4 τ + {E 4 τ} 2 2E 2 τe 6 τ 9 into the equation 4.2, one can easily see that f 2 τ is a cusp form for the function f τ given by f τ = G 4 τ, that is, we have cf. 3, p. 90, Exercise.20 c and 2 f 2 τ = τ 2 4 3 3 5 2 π 2. Similarly, if f 2 τ is a cusp form for the function f τ given by f τ = E 4 τ, then we can readily find that cf. 3, p. 90, Exercise.20 c and 2 f 2 τ = 5π2 3888 τ. Thus, if f τ is a modular form, then f 2 τ is a cusp form cf. 3, p. 90, Exercise.20 b. If we set f τ = E 6 τ in the equation 4., we have d 2 f 2 τ = 7 dτ {E 6τ} 6E 6 τ d2 dτ {E 6τ}. 4.3 2 By substituting the following well-known derivative formulas for E 6 τ cf. 6, 0 and 2: d dτ {E 6τ} = πi E 2 τe 6 τ {E 4 τ} 2 and cf. 6 d 2 dτ {E 6τ} = 7π2 {E2 τ} 2 E 2 6 τ + E 4 τe 6 τ 2E 2 τ {E 4 τ} 2 6 into the equation 4.3, we arrive at the following results. Corollary 2. It is asserted that f 2 τ = 7π 2 {E 6 τ} 2 E 4 τ {E 4 τ} 4. Corollary 3. Let n, r and k \ {, 2}. Suppose also that If a 0 and \ {}. 2m + r = 2 n k + 42 n, then the sequence { r 2m } a T z ; w, m which is a modular form with weight 2m + r, satisfies Theorem 6.

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