EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 4, 2022, 1662-1682 ISSN 1307-5543 – ejpam.com Published by New York Business Global Fourier Series for Bernoulli-Type Polynomials, Euler-Type Polynomials and Genocchi-Type Polynomials of Integer Order Cristina B. Corcino1,2, Roberto B. Corcino1,2,∗ 1 Research Institute for Computational Mathematics and Physics, Cebu Normal University, 6000 Cebu City, Philippines 2 Department of Mathematics, Cebu Normal University, 6000 Cebu City, Philippines Abstract. Parameters a, b, c, and α are introduced to form the Bernoulli-type, Euler-type and Genocchi-type polynomilas where α is the order of the polynomial and is a positive integer. Ana- lytic methods are used here to obtain the Fourier series for these polynomials. 2020 Mathematics Subject Classifications: 11B68, 42A16, 11M35 Key Words and Phrases: Fourier Series, Bernoulli polynomials, Euler polynomials, Genocchi polynomials 1. Introduction The polynomials that will be considered are given by the generating functions (1)-(3) where B (α) n (x; a, b, c) denotes the Bernoulli-type polynomials of order α, E (α) n (x; a, b, c) denotes the Euler-type polynomials of order α and G (α) n (x; a, b, c) denotes the Genocchi- type polynomials of order α with α ∈ Z+, a, b, c are positive real numbers and B = ln b− ln a > 0. ( t bt − at )α cxt = ∞∑ n=0 B(α) n (x; a, b, c) tn n! , |t| < 2π B (1) ( 2 bt + at )α cxt = ∞∑ n=0 E(α) n (x; a, b, c) tn n! , |t| < π B (2) ( 2t bt + at )α cxt = ∞∑ n=0 G(α) n (x; a, b, c) tn n! , |t| < π B . (3) ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4507 Email addresses: corcinoc@cnu.edu.ph (C. Corcino), rcorcino@yahoo.com (R. Corcino) https://www.ejpam.com 1662 © 2022 EJPAM All rights reserved. C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1662-1682 1663 These polynomials are generalizations of the classical Bernoulli, Euler and Genocchi poly- nomials, respectively. The Apostol-type of these polynomials were mentioned in [9] in the introduction of the paper. Fourier series for the tangent type of these polynomials were obtained in [7] while the Fourier series for the Apostol-Tangent polynomials were obtained in [6]. Integral representation and explicit formula at rational arguments of tangent poly- nomials of higher order were derived in [8]. Properties of higher order Apostol-Frobenius- type poly-Genocchi polynomials with parameters a, b and c were studied in [10]. Other interesting polynomials related to Bernoulli, Euler and Genocchi were studied in [1–4]. In this paper, the Fourier series for B (α) n (x; a, b, c), E (α) n (x; a, b, c) and G (α) n (x; a, b, c) of positive integer order α will be derived. The method used here is analytic. In particular, there will be heavy use of contour integration and residue theory. For elaborate discussion of these topics see [5]. 2. The case α = 1 Lemma 2.1. Let n ≥ 2, N > 1 and CN be the circle about zero of radius R = (2Nπ−ε)/B, where 0 < ε < 1 and B = ln b− ln a, b > a. For 0 < x < ( ln a− B 2π − ε )/ ln c, ln c > 0 we have lim N→+∞ ∫ CN cxt bt − at dt tn = 0. Proof. ∣∣∣∣∫ CN cxt (bt − at) dt tn ∣∣∣∣ ≤ ∫ CN |cxt| |bt − at| |dt| |tn| . We will show that under the conditions in the lemma, the function cxt (bt − at) is bounded on CN . Write cxt = ext ln c, bt = et ln b, at = et ln a, where t ∈ CN . Let t = γ + iρ. Then γ = 2Nπ − ε B cos θ, ρ = 2Nπ − ε b sin θ, where 0 ≤ θ ≤ 2π. Then |cxt| |bt − at| = exγ ln c |e(γ+iρ) ln b − e(γ+iρ) ln a| = exγ ln c eγ ln a[e2γB − 2eγB cos ρB + 1] 1 2 C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1662-1682 1664 = 1 eγ[ln a−x ln c][e2γB − 2eγB cos ρB + 1] 1 2 . With x < ln a ln c − B (2π − ε) ln c =⇒ x ln c < ln a− B 2π − ε =⇒ x ln c− ln a < − B 2π − ε =⇒ ln a− x ln c > B 2π − ε ≥ B 2πN − ε , ∀ N ≥ 1. Thus, 1 eγ[ln a−x ln c] ≤ 1 ecosθ ≤ 1 e−1 = e, and |cxt| |bt − at| ≤ e [e2γB − 2eγB cos ρB + 1] 1 2 . The denominator of the preceding expression must not be zero. With 0 ≤ θ ≤ 2π, we look at 3 cases: Case 1: cos θ < 0 As N → +∞, γ → −∞ and e2γB − 2eγB cos ρB + 1 −→ 1 provided B > 0. Case 2: cos θ > 0 As N → +∞, γ → +∞ and e2γB − 2eγB cos ρB+1 = e2γB ( 1− 2 cos ρB eγB + 1 eγB ) −→ +∞, provided B > 0. Case 3: cos θ = 0 Then γ = 0 and e2γB − 2eγB cos ρB+1 = 2− 2 cos ρB, which is nonzero provided that cos ρB ̸= 1. Because cos θ = 0, we have ρ = ±(2Nπ − ε)/B. Thus, cos ρB = cos[(±2Nπ − ε)] = 1 iff 2Nπ − ε = 2kπ, for some integer k. This gives 2(N − k)π = ε, C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1662-1682 1665 which is not possible because 0 < ε < 1. Thus, under the conditions in the lemma, in all 3 cases cxt/(bt − at) is bounded ∀t ∈ CN . Let M be a positive integer such that∣∣∣∣ cxt bt − at ∣∣∣∣ < M . Then ∣∣∣∣∫ CN cxt bt − at dt tn ∣∣∣∣ < M ∫ CN |dt| |tn| = M · (2Nπ − ε)2π (2Nπ − ε)n Bn−1 = 2MπBn−1 (2Nπ − ε)n−1 −→ 0 as N → +∞ for n ≥ 2. This completes the proof of the lemma. Theorem 2.2. Let a, b, c be positive real numbers. The Fourier series of the Bernoulli-type polynomials Bn(x; a, b, c) is given by Bn(x; a, b, c) n! = − 1 B ∑ k∈Z+ etk(x ln c−ln a) tnk , valid for 0 < x < ( ln a− B 2π − ε )/ ln c, ln c > 0 where tk = 2kπi/B, B = ln b− ln a > 0. Proof. When α = 1, the generating function (1) reduces to t bt − at cxt = ∞∑ n=0 Bn(x; a, b, c) tn n! , |t| < 2π B . Applying the Cauchy Integral Formula yields Bn(x; a, b, c) n! = 1 2πi ∫ C cxt bt − at dt tn , where C is a circle with center at 0 and radius less than 2π B . Let f(t) = cxt (bt − at)tn . C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1662-1682 1666 The function f(t) has simple poles at t such that bt − at = 0 and a pole at t = 0 of order n. Let tk be those values of t such that bt − at = 0. These values are obtained as follows. bt − at = 0 et ln b − et ln a = 0 (et ln b = et ln a)e−t ln a log(et(ln b−ln a) = 1) t(ln b− ln a) = log 1 = i Arg 1 + 2kπi t = 2kπi B , where B = ln b− ln a. Let tk = 2kπi/B, k ∈ Z. Now let CN be the circle described in Lemma 2.1. Applying the Residue Theorem, we have lim N→+∞ 1 2πi ∫ CN cxt bt − at dt tn = Res(f(t), t = 0) + ∑ k∈Z,k ̸=0 Res(f(t), t = tk). By Lemma 2.1, 0 =Res(f(t), t = 0) + ∑ k∈Z,k ̸=0 Res(f(t), t = tk) 0 = Bn(x; a, b, c) n! + ∑ k∈Z,k ̸=0 Res(f(t), t = tk) =⇒Bn(x; a, b, c) n! = − ∑ k∈Z,k ̸=0 Res(f(t), t = tk). Computing the residue at tk: Res(f(t), t = tk) = lim t→tk (t− tk) 2cxt (bt − at)tn = 2cxtkt−n k µ , where µ = d dt (bt − at)|t=tk = d dt (et ln b − et ln a)|t=tk = (ln b etk ln b − ln a etk ln a) e−tk ln a e−tk ln a C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1662-1682 1667 = etk ln a(ln b etk(ln b−ln a) − ln a) = etk ln a(ln b− ln a) = B · etk ln a. Thus, Res(f(t), t = tk) = cxtkt−n k B · etk ln a = etk(x ln c−ln a) B · tnk . Consequently, Bn(x; a, b, c) n! = − 1 B ∑ k∈Z,k ̸=0 etk(x ln c−ln a) tnk . Lemma 2.3. Let a, b, c be positive real numbers. Let n ≥ 1, N > 1 and CN be the circle about zero of radius R = ((2N + 1)π − ε)/B, where 0 < ε < 1 and B = ln b− ln a, b > a. For 0 < x < ( ln a− B π − ε )/ ln c, ln c > 0 we have lim N→+∞ ∫ CN cxt bt + at dt tn+1 = 0. Proof. We will show that the function cxt bt + at is bounded on CN under the conditions in Lemma 2.3. From the proof of Lemma 2.1, |cxt| |bt + at| = exγ ln c eγ ln a[e2γB + 2eγB cos ρB + 1] 1 2 , where here, γ = (2N + 1)π − ε B cos θ, ρ = (2N + 1)π − ε B sin θ, 0 ≤ θ ≤ 2π. With x < ln a ln c − B (π − ε) ln c =⇒ ln a− x ln c > B π − ε ≥ B (2N + 1)π − ε , ∀ N ≥ 0. C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1662-1682 1668 Then 1 eγ(ln a−x ln c) ≤ 1 ecos θ ≤ 1 e−1 = e. Thus, |cxt| |bt + at| ≤ e [e2γB + 2eγB cos ρB + 1] 1 2 . The expression e2γB+2eγB cos ρB+1 must not be zero. The results for the cases cos θ < 0 and cos θ > 0 obtained in the proof of Lemma 2.1 still hold. We reconsider here the case cos θ = 0. In the case θ = 0, γ = 0 and e2γB + 2eγB cos ρB + 1 = 2 + 2 cos ρB, which is nonzero provided that cos ρB ̸= −1. Since cos θ = 0, we have ρ = (±1) (2N + 1)π − ε B . Thus, cos ρB = cos(±(2N + 1)π − ε) = −1 iff (2N + 1)π − ε = (2k + 1)π, for some integer k. Equivalently, (2N + 1)π − (2k + 1)π = ε 2(N − k)π = ε, which is not possible because 0 < ε < 1. Thus, under the conditions in the Lemma, the function cxt bt + at is bounded on CN as N → +∞. Let M∗ be a positive integer such that |cxt| |bt + at| < M∗, ∀t ∈ CN . Then ∣∣∣∣∫ CN cxt bt + at · dt tn+1 ∣∣∣∣ ≤ ∫ CN ∣∣∣∣ cxt bt + at ∣∣∣∣ |dt| |tn+1| < M∗ (2N + 1)π − ε B · 2π( (2N + 1)π − ε B )n+1 < 2M∗πBn ((2N + 1)π − ε)n , which goes to zero as N → +∞. C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1662-1682 1669 Theorem 2.4. Let a, b, c be positive real numbers. The Fourier series of the Euler-type polynomials En(x; a, b, c) is given by En(x; a, b, c) n! = 2 B ∑ k∈Z etk(x ln c−ln a) tn+1 k , valid for 0 < x < ( ln a− B π − ε )/ ln c, ln c > 0 where tk = (2k + 1)πi/B, B = ln b− ln a > 0. Proof. When α = 1, the generating function (2) reduces to( 2 bt + at ) cxt = ∞∑ n=0 En(x; a, b, c) tn n! , |t| < π B . Applying the Cauchy Integral Formula, En(x; a, b, c) n! = 1 2πi ∫ C 2cxt (bt + at)tn+1 dt , where C is a circle about zero of radius π B . Let g(t) = 2cxt (bt + at)tn+1 . The function g(t) has a pole at t = 0 of order n+1 and simple poles at the values of t such that bt + at = 0. These values are tk = (2k + 1)πi/B, k ∈ Z which are obtained similarly as those in Theorem 2.2. Let CN be the circle described in Lemma 2.3. From the Residue Theorem, lim N→+∞ 1 2πi ∫ CN g(t)d(t) = Res(g(t), t = 0) + ∑ k∈Z Res(g(t), t = tk). By Lemma 2.3, we have En(x; a, b, c) n! = − ∑ k∈Z Res(g(t), t = tk). Computing the residues of g(t) at tk: Res(g(t), t = tk) = lim t→tk (t− tk) 2ext ln c bt + at t−n−1 = 2extk ln ct−n−1 k ν , C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1662-1682 1670 where ν = d dt (bt + at)|t=tk = ((ln b)etk ln b + (ln a)etk ln a) e−tk ln a e−tk ln a = etk ln a[(ln b)etk(ln b−ln a) + ln a] = etk ln a[− ln b+ ln a] = −B · etk ln a. Thus, Res(g(t), t = tk) = 2etkx ln ct−n−1 k −B · etk ln a = 2etk(x ln c−ln a) −B · tn+1 k . Consequently, En(x; a, b, c) n! = 2 B ∑ k∈Z etk(x ln c−ln a) tn+1 k . Theorem 2.5. Let a, b, c be positive real numbers. The Fourier series of the Genocchi-type polynomials Gn(x; a, b, c) is given by Gn(x; a, b, c) n! = 2 B ∑ k∈Z etk(x ln c−ln a) tnk , valid for 0 < x < ( ln a− B π − ε )/ ln c, ln c > 0 where tk = (2k + 1)πi/B, B = ln b− ln a > 0. Proof. The theorem follows from Theorem 2.4. 3. The case α ≥ 2 Lemma 3.1. Let a, b, c be positive real numbers. Let n ≥ α ≥ 2, α ∈ Z+, N > 1 and CN be the circle about zero of radius R = (2Nπ− ε)/B, where 0 < ε < 1 and B = ln b− ln a > 0. For 0 < x < ( α ln a− B 2π − ε )/ ln c, ln c > 0 we have lim N→+∞ ∫ CN cxt (bt − at)α dt tn−α+1 = 0. C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1662-1682 1671 Proof. We will show that the function cxt (bt − at)α is bounded on CN . From Lemma 2.1, |bt − at| = eγ ln a[e2γB − 2eγB cos ρB + 1] 1 2 , where t ∈ CN , t = γ + iρ. That is, γ = 2Nπ − ε B cos θ, ρ = 2Nπ − ε B sin θ, 0 ≤ θ ≤ 2π. Then |bt − at|α = eαγ ln a[e2γB − 2eγB cos ρB + 1] α 2 , and ∣∣∣∣ cxt (bt − at)α ∣∣∣∣ = exγ ln c eαγ ln a[e2γB − 2eγBcosρB + 1] α 2 . Impose that α ln a− x ln c > B 2Nπ − ε , ∀N ≥ 1. This is satisfied when α ln a− x ln c > B 2π − ε . Equivalently, impose that 0 < x < ( α ln a− B 2π − ε )/ ln c. Then 1 eγ(α ln a−x ln c) < 1 ecos θ ≤ 1 e−1 = e. Consequently, ∣∣∣∣ cxt (bt − at)α ∣∣∣∣ < e [e2γB − 2eγB cos ρB + 1] α 2 . It follows from Lemma 2.1 that the right hand side above is bounded on CN as N → +∞. That is, there is a constant M such that∣∣∣∣ cxt (bt − at)α ∣∣∣∣ < M, t ∈ CN and 0 < x < ( α ln a− B 2π − ε )/ ln c. Thus, ∣∣∣∣∫ CN cxt (bt − at)α dt tn−α+1 ∣∣∣∣ < M ∫ CN |dt| |tn−α+1| < M · 2Nπ − ε B · 2π( 2Nπ − ε B )n−α+1 C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1662-1682 1672 = 2πMBα−n (2Nπ − ε)n−α , n ≥ α. −→ 0 as N → +∞. Lemma 3.2. For a, b, c ∈ R+, x ∈ R, ν, α ∈ Z+ with fixed ν ≥ α, B(α) ν (x; a, b, c) = ν∑ l=0 ( ν l ) B (α) l (0; a, b, c)(x ln c)ν−l. Proof. ( t bt − at )α cxt · cyt = ( ∞∑ n=0 B(α) n (x; a, b, c) tn n! )( ∞∑ n=0 (yt ln c)n n! ) ( t bt − at )α c(x+y)t = ∞∑ n=0 n∑ l=0 B (α) l (x; a, b, c) tl l! (yt ln c)n−l (n− l)! · n! n! ∞∑ n=0 B(α) n (x+ y; a, b, c) ty n! = ∞∑ n=0 n∑ l=0 ( n l ) B (α) l (x; a, b, c)(y ln c)n−l t n n! . Thus, B(α) n (x+ y; a, b, c) = n∑ l=0 ( n l ) B (α) l (x; a, b, c)(y ln c)n−l. Take y = z, x = 0. Then Bα n (z; a, b, c) = n∑ l=0 ( n l ) B (α) l (0; a, b, c)(z ln c)n−l Now take n = ν and z = x, we have B(α) ν (x; a, b, c) = ν∑ l=0 ( ν l ) B (α) l (0; a, b, c)(x ln c)v−l. Theorem 3.3. Let a, b, c be positive real numbers, N,n, α ∈ Z+ with n ≥ α ≥ 2, N > 1 and CN be the circle about zero of radius R = (2Nπ − ε)/B, where 0 < ε < 1 and B = ln b − ln a > 0. The Fourier series of the Bernoulli-type polynomials B (α) n (x; a, b, c) of order α is given by C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1662-1682 1673 B (α) n (x; a, b, c) n! = − ∑ k∈Z,k ̸=0 ( α−1∑ ν=0 (α− n− 1)α−1−ν ν!(α− 1− ν)! (2kπi)νB(α) ν (x; a, b, c) ) e2kπi(x ln c−α ln b) (2kπi)n , valid for 0 < x < ( α ln a− B 2π − ε )/ ln c, ln c > 0, where B (α) ν (x; a, b, c) is given in Lemma 3.2. Proof. Applying the Cauchy Integral Formula to (1), B (α) n (x; a, b, c) n! = 1 2πi ∫ C cxt (bt − at)α dt tn+1−α , where C is a circle about the origin with radius less than 2π B . Let fα(t) = cxt (bt − at)αtn−α+1 , n > α. The function fα(t) has a pole of order n − α + 1 at t = 0 and a pole of order α at the zeros of bt − at which are given by tk = 2kπi B , k ∈ Z. Now let CN , N > 1 be the circle described in Lemma 3.1. Applying the Residue Theorem, lim N→+∞ 1 2πi ∫ CN cxt (bt − at)α dt tn−α+1 = Res(fα(t), t = 0) + ∑ k∈Z,k ̸=0 Res(fα(t), t = tk). By Lemma 3.1, 0 = Res(fα(t), t = 0) + ∑ k∈Z,k ̸=0 Res(fα(t), t = tk) 0 = B (α) n (x; a, b, c) n! + ∑ k∈Z,k ̸=0 Res(fα(t), t = tk) ⇐⇒ Bα n (x; a, b, c) n! = − ∑ k∈Z,k ̸=0 Res(fα(t), t = tk). (4) Computing the residues at tk: Res(fα(t), t = k) = 1 (α− 1)! lim t→tk dα−1 dtα−1 (t− tk) α ( ext ln c (bt − at)α ) 1 tn−α+1 = 1 (α− 1)! lim t→tk dα−1 dtα−1 [ (t− tk) α (bt − at)α ext ln c tn−α+1 ] . (5) C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1662-1682 1674 Taking x = 0 in (1) gives ( t bt − at )α = ∞∑ n=0 B(α) n (0; a, b, c) tn n! . Replacing t 7→ t− tk and writing bt = et ln b, at = et ln a, (t− tk) α (e(t−tk) ln b − e(t−tk) ln a)α = ∞∑ n=0 B(α) n (0; a, b, c) (t− tk) n n! . (6) Multiplying and dividing the left hand side of (6) by eαtk ln b gives (t− tk) αeαtk ln b (et ln b − et ln aetkB)α = ∞∑ n=0 B(α) n (0; a, b, c) (t− tk) n n! . (7) With tk = (2kπi)/B, we have etkB = e2kπi = 1. Thus, (7) becomes (t− tk) αeαtk ln b (bt − at)α = ∞∑ n=0 B(α) n (0; a, b, c) (t− tk) n n! (t− tk) α (bt − at)α = e−αtk ln b ∞∑ n=0 B(α) n (0; a, b, c) (t− tk) n n! . (8) Substituting (8) to (5) gives, Res(fα(t), t = tk) = e−αtk ln b (α− 1)! lim t→tk dα−1 dtα−1 ( ext ln c tn−α+1 ∞∑ n=0 Bn(0; a, b, c) (t− tk) n n! ) . The derivatives will be obtained using Leibniz Rule. This is done as follows. Recalling the Leibniz Rule for derivatives, dn dtn (fg) = n∑ k=0 ( n k )( dn−k dtn−k f )( dk dtk g ) . Let f = tα−n−1, g = ext ln c ∑∞ n=0B (α) n (0; a, b, c) (t− tk) n n! . Then dα−1 dtα−1 (fg) = α−1∑ ν=0 ( α− 1 ν )( dd−1−ν dtd−1−ν f )( dν dtν g ) C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1662-1682 1675 = α−1∑ ν=0 ( α− 1 ν ) (α− n− 1)α−1−νt α−n−1−(α−1−ν) ( dν dtν g ) = α−1∑ ν=0 ( α− 1 ν ) (α− n− 1)α−1−νt −n+ν ( dν dtν g ) , (9) where the notation (n)k is designed as (n)k = n(n− 1)(n− 2)...(n− k + 1). Also, (α− n− 1)α−1−ν = (−1)α−1−ν(n− α+ 1)(n− α+ 2)(n− α+ 3)... ((n− α) + α− ν − 1) = (−1)α−1−ν ⟨n− α+ 1⟩α−ν−1 . On the other hand, dν dtν g = dν dtν ( ∞∑ n=0 B(α) n (0; a, b, c) (t− tk) n n! · ext ln c ) = ν∑ l=0 ( ν l ) dν−l dtv−l et(x ln c) · dl dtl ( ∞∑ n=0 B(α) n (0; a, b, c) (t− tk) n n! ) = ν∑ l=0 ( ν l ) (x ln c)ν−lext ln c ∑ n≥l B(α) n (0; a, b, c)(n)l (t− tk) n−l n! . Now take the limit as t → tk. Then lim t→tk dν dtν g = ν∑ l=0 ( ν l ) (x ln c)v−letkx ln cB (α) l (0; a, b, c). Substituting to (9) and taking the limit as t → tk will yield lim t→k dα−1 dtα−1 (fg) = α−1∑ ν=0 ( α− 1 ν ) (α− n− 1)α−1−ν t−n+ν k ν∑ l=0 ( ν l ) (x ln c)ν−letk ln cB (α) l (0; a, b, c) = α−1∑ ν=0 ( α− 1 ν ) (α− n− 1)α−1−ν t−n+ν k etk ln c ( ν∑ l=0 ( ν l ) (x ln c)v−lB (α) l (0; a, b, c) ) . (10) C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1662-1682 1676 Applying Lemma 3.2 to (10), lim t→k dα−1 dtα−1 (fg) = α−1∑ ν=0 ( α− 1 ν ) (α− n− 1)α−1−νt −n+ν k etk ln cB(α) ν (x; a, b, c). Thus, Res(fα(t), t = tk) = etk(x ln c−α ln b) tnk α−1∑ ν=0 (α− n− 1)α−1−ν ν!(α− 1− ν)! tνkB (α) ν (x; a, b, c). (11) The desired Fourier series is obtained by substituting (11) to (4). Taking α = 1, the Fourier series in Theorem 3.3 reduces to that in Theorem 2.2. For α = 2, Theorem 3.3 gives the Fourier series of the Bernoulli-type polynomials of order 2. This is given by B (2) n (x; a, b, c) n! = −1 B2 ∑ k∈Z,k ̸=0 (−n+ 1 + x ln c) e2kπi(x ln c−2 ln b) (2kπi)n , valid under the conditions in Theorem 3.3. Lemma 3.4. Let a, b, c be positive real numbers with b > a, n, α ∈ Z+ with n ≥ α, N > 1 and CN be the circle about zero of radius R = ((2N + 1)π − ε)/B, where 0 < ε < 1 and B = ln b− ln a. For ln c > 0 and 0 < x < ( α ln a− B π − ε )/ ln c (12) we have lim N→+∞ ∫ CN cxt (bt + at)α dt tn+1 = 0. Proof. From the proof of Lemma 3.2, |bt + at|α = eαγ ln a[e2γB + 2eγB cos ρB + 1] α 2 , t ∈ CN where t = γ + iρ = (2N + 1)π − ε B (cos θ + i sin θ), 0 ≤ θ ≤ 2π. Thus, γ = (2N + 1)π − ε B cos θ, ρ = (2N + 1)π − ε B sin θ. C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1662-1682 1677 For x satisfying (12), it follows that α ln a− x ln c > B π − ε ≥ B (2N + 1)π − ε , ∀N ≥ 1. Then 1 eγ[α ln a−x ln c] = 1 e (2N+1)−ε B cos θ[α ln a−x ln c] < 1 ecos θ < e. Consequently,∣∣∣∣ cxt (bt + at)α ∣∣∣∣ = |cxt| |bt + at|α = 1 eγ[α ln a−x ln c][e2γB + 2eγB cos ρB + 1] α 2 < e (e2γB + 2eγB cos ρB + 1) α 2 . The expression e2γB + 2eγB cos ρB + 1 ̸= 0 ∀t ∈ CN as discussed in Lemma 2.3. Thus, ∃ an integer M s.t. ∣∣∣∣ cxt (bt + at)α ∣∣∣∣ < M, ∀t ∈ CN . Hence, ∣∣∣∣∫ CN cxt (bt + at)α dt tn+1 ∣∣∣∣ ≤ M ∫ CN |dt| |tn+1| = M · (2N + 1)π − ε B · 2π( (2N + 1)π − ε B )n+1 = 2πMBn ((2N + 1)π − ε)n+1 , n > 1. −→ 0 as N → +∞. Lemma 3.5. For a, b, c ∈ R+, x ∈ R, ν, α ∈ Z+ with fixed ν ≥ α ≥ 2, E(α) ν (x; a, b, c) = ν∑ l=0 ( ν l ) E (α) l (0; a, b, c)(x ln c)v−l. Proof. The proof is done similarly as that of Lemma 3.2. Theorem 3.6. Let a, b, c be positive real numbers with b > a, N , n, α ∈ Z+, n ≥ α ≥ 2, N > 1 and CN be the circle about zero of radius R = ((2N +1)π− ε)/B, where 0 < ε < 1 C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1662-1682 1678 and B = ln b − ln a. The Fourier series of the Euler-type polynomials E (α) n (x; a, b, c) of order α is given by E (α) n (x; a, b, c) n! = −2α (α− 1)! ∑ k∈Z α−1∑ ν=0 ( α− 1 ν ) (−n− 1)α−1−νB (α) ν (x; a, b, c) etk(x ln c−α ln b) tn+α−ν k , valid for 0 < x < ( α ln a− B π − ε )/ ln c, ln c > 0. Proof. Applying the Cauchy-Integral Formula to (2), E (α) n (x; a, b, c) n! = 1 2πi ∫ C 2αcxt (bt + at)α dt tn+1 , where C is a circle about zero of radius less than π B . Let gα(t) = cxt (bt + at)αtn+1 . Then E (α) n (x; a, b, c) 2α(n!) = 1 2πi ∫ C gα(t)dt. The function gα(t) has a pole of order n + 1 at t = 0 and a pole of order α at the zeros of bt + at which are given by tk = ((2k + 1)πi)/B, k ∈ Z. Applying the Residue Theorem and taking the limit as N → +∞, lim N→+∞ 1 2πi ∫ C gα(t)dt = Res(gα(t), t = 0) + ∑ k∈Z Res(gα(t), t = tk). It follows from Lemma 3.4 that E (α) n (x; a, b, c) 2α(n!) = − ∑ k∈Z Res(gα(t), t = tk). Computing the residues at tk: Res(gα(t), t = tk) = 1 (α− 1)! lim t→tk dα−1 dtα−1 ( (t− tk) α cxt (bt + at)α · 1 tn+1 ) . (13) Now use (7). With tk = (2k + 1)πi/B, etkB = e(2k+1)πi = −1. Thus, (7) becomes, (t− tk) αeαtk ln b (et ln b + et ln a)α = ∞∑ n=0 B(α) n (0; a, b, c) (t− tk) n n! C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1662-1682 1679 (t− tk) α (bt + at)α = e−αtk ln b ∞∑ n=0 B(α) n (0; a, b, c) (t− tk) n n! . (14) Substituting (14) to (13), Res(gα(t), t = tk) = e−αtk ln b (α− 1)! lim t→tk dα−1 dtα−1 ( cxtt−n−1 ∞∑ n=0 B(α) n (0; a, b, c) (t− tk) n n! ) . Applying the Leibniz Rule for differentiation, Res(gα(t), t = tk) = etk(x ln c−α ln b) (α− 1)! α−1∑ ν=0 ( α− 1 ν ) (−n− 1)α−1−ν t−n−α+ν k B(α) ν (x; a, b, c). Thus, E (α) n (x; a, b, c) n! = − 2α (α− 1)! ∑ k∈Z etk(x ln c−α ln b) α−1∑ ν=0 ( α− 1 ν ) (−n− 1)α−1−ν t−n−α+ν k B(α) ν (x; a, b, c) = − 2α (α− 1)! ∑ k∈Z α−1∑ ν=0 ( α− 1 ν ) (−n− 1)α−1−ν B(α) ν (x; a, b, c) etk(x ln c−α ln b) tn+α−ν k , which is the desired Fourier series of E (α) n (x; a, b, c). Taking α = 1, the Fourier series in Theorem 3.6 reduces to that in Theorem 2.4. For α = 2, the Fourier series is given by E (2) n (x; a, b, c) 22(n!) = − ∑ k∈Z (−n− 1)B (2) 0 (x; a, b, c) etk(x ln c−2 ln b) tn+2 k +B (2) 1 (x; a, b, c) etk(x ln c−2 ln b) tn+1 k , where B (2) 0 (x; a, b, c) = 1 B2 , (15) B (2) 1 (x; a, b, c) = x ln c B2 + ln ab− (ln b)2 − ln b ln a− (ln a)2 B2 . (16) Lemma 3.7. Let a, b, c be positive real numbers with b > a. Let N,n, α ∈ Z+, N > 1 and CN be the circle about zero with raidus R = (2N + 1)π − ε B , where 0 < ε < 1 and B = ln b− ln a. For 0 < x < ( α ln a− B π − ε )/ ln c, ln c > 0 we have lim N→+∞ ∫ CN cxt (bt + at)α dt tn−α+1 = 0. C. Corcino, R. Corcino / Eur. J. Pure Appl. Math, 15 (4) (2022), 1662-1682 1680 Proof. This follows from Lemma 3.4. Lemma 3.8. For a, b, c ∈ R+, x ∈ R, ν, α ∈ Z+ with fixed ν ≥ α, G(α) ν (x; a, b, c) = ν∑ l=0 ( ν l ) G (α) l (0; a, b, c)(x ln c)ν−l. Proof. The proof is done similarly as that of Lemma 3.2. Theorem 3.9. Let a, b, c be positive real numbers with b > a. Let N,n, α ∈ Z+ with n ≥ α ≥ 2, N > 1 and CN be the circle about zero of radius R = ((2N + 1)π − ε)/B, where 0 < ε < 1 and B = ln b− ln a. The Fourier series of the Genocchi-type polynomials G (α) n (x; a, b, c) of order α is given by G (α) n (x; a, b, c) n! = − 2α (α− 1)! ∑ k∈Z α−1∑ ν=0 ( α− 1 ν ) (α−n−1)α−1−ν B(α) ν (x; a, b, c) etk(x ln c−α ln b) tn−ν k . Proof. Applying the Cauchy Integral Formula to (3), G (α) n (x; a, b, c) n! = 2α 2πi ∫ C cxt (bt + at)α dt tn−α+1 , where C is a circle about zero of radius < π/B. Let hα(t) = cxt (bt + at)αtn−α+1 . This function has a pole of order n − α + 1 at t = 0 and a pole of order α at the zeros of bt + at. These poles are given by tk = (2k + 1)πi/B, k ∈ Z. Applying the Residue Theorem and taking the limit as N → +∞, lim N→+∞ 1 2πi ∫ C hα(t)dt = Res(hα(t), t = 0) + ∑ k∈Z Res(hα(t), t = tk). It follows from Lemma 3.7 that G (α) n (x; a, b, c) n! 2α = − ∑ k∈Z Res(hα(t), t = tk), (17) where Res(hα(t), t = tk) = 1 (α− 1)! lim t→tk dα−1 dtα−1 ( (t− tk) α cxt (bt + at)α · 1 tn+1−α ) . From (14), Res(hα(t), t = tk) = e−αtk ln b (α− 1)! lim t→tk dα−1 dtα−1 ( cxtt−n+α−1 ∞∑ n=0 B(α) n (0; a, b, c) (t− tk) α n! ) . REFERENCES 1681 Following the computation in the Euler-type polynomials, Res(hα(t), t = tk) = etk(x ln c−α ln b) (α− 1)! α−1∑ ν=0 ( α− 1 ν ) (α− n− 1)α−1−ν t−n+ν k B(α) ν (x; a, b, c). (18) Substituting (18) to (17) gives the desired Fourier series. Taking α = 1, the Fourier series in Theorem 3.9 reduces to that in Theorem 2.5. Taking α = 2 and n = 4, the series gives G (2) 4 (x; a, b, c) 22(4!) = − ∑ k∈Z { −3B (2) 0 (x; a, b, c) e(2k+1)πi(x ln c−2 ln b) ((2k + 1)πi)4 + B (2) 1 (x; a, b, c) e(2k+1)πi(x ln c−2 ln b) ((2k + 1)πi)3 } where B (2) 0 (x; a, b, c) and B (2) 1 (x; a, b, c) are given in (15) and (16), respectively. 4. Some Remarks The Fourier series expansions obtained in this paper for B (α) n (x; a, b, c), E (α) n (x; a, b, c) and G (α) n (x; a, b, c) are useful in establishing the asymptotic formulas of these polynomials. It would then be interesting to investigate the asymptotic behavior of these polynomials. Acknowledgements This research is funded by Cebu Normal University through its Center for Research and Development and the Research Institute for Computational Mathematics and Physics. References [1] Bedoya, D., Ortega, M., Ramirez, W., Urieles, A., Fourier expansion and integral representation generalized Apostol-type Frobenius-Euler polynomials. Adv. Differ. Equ. 2020 (2020), Article 534. [2] Bedoya, D., Ortega, M., Ramirez, W., Urieles. New biparametric families of Apostol- Frobenius-Euler polynomials of level m, Mat. Stud. 55 (2021), 10-23. [3] Cesarano, C., Ramirez, W., Khan, S. A new class of degenerate Apostol?type Hermite polynomials and applications, Dolomites Res. Notes Approx. 15 (2022), 1-10. [4] Cesarano, C., Ramirez, W., Some new classes of degenerated generalized Apostol- Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, Carpathian Math. Publ. 14(2) (2022). REFERENCES 1682 [5] Churchill, R. V., Brown, J. W. Complex Variable and Applications, 8th ed.; McGraw- Hill Book: New York, NY, USA, 2008. [6] Corcino, C., Castañeda, W., Corcino, R., Asymptotic Approximations of Apostol- Tangent Polynomials in terms of Hyperbolic Functions, Computer Modeling in Engi- neering and Sciences, 132(1) (2022), 133-151. [7] Corcino, C., Corcino, R., Fourier series for the tangent polynomials, tangent-Bernoulli and tangent-Genocchi polynomials of higher order, Axioms 11(3) (2022), Article 86. [8] Corcino, C., Corcino, R., Casquejo, J.; Fourier expansion,integral representation and explicit formula at rational arguments of the tangent polynomials of higher-order, European Journal of Pure and Applied Mathematics 14(4) (2021), 1457-1466. [9] Khan, W., Srivastava, D., On the generalized Apostol-type Frobenius-Genocchi poly- nomials, Filomat 33(7) (2019), 1967-1977. [10] Corcino, R., Corcino, C. Higher Order Apostol-Frobenius-Type Poly-Genocchi Poly- nomials With Parameters a, b and c. J. Inequal. Spec. Funct. 12 (2021), 54-72.