Abstract
Fourier expansions of higher-order Apostol–Genocchi and Apostol–Bernoulli polynomials are obtained using Laurent series and residues. The Fourier expansion of higher-order Apostol–Euler polynomials is obtained as a consequence.
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1 Introduction
Higher-order Apostol–Genocchi, Apostol–Bernoulli, and Apostol–Euler polynomials are defined by the following relations, respectively (see [7]):
When \(m=1\), the above equations give the generating functions for the Apostol–Genocchi, Apostol–Bernoulli, and Apostol–Euler polynomials, respectively (see [3]). When \(m=1\) and \(\lambda=1\), the equations give the generating functions for the classical Genocchi, Bernoulli, and Euler polynomials (see [4, 10]).
New formulas for the product of an arbitrary number of the Apostol–Bernoulli, Apostol–Euler, and Apostol–Genocchi polynomials were established in [6] where these polynomials were referred to as Apostol-type polynomials. Further, higher-order convolutions for these polynomials were established in [7]. New identities for the Apostol–Bernoulli polynomials and Apostol–Genocchi polynomials were also presented in [8].
Fourier expansion, being a sum of multiple of sines and cosines, is easily differentiated and integrated, which often simplifies analysis of functions such as saw waves which are common signals in experimentation [9]. Real world applications of Fourier series include the use for audio compression [5].
Fourier expansions of Genocchi polynomials and Apostol–Genocchi polynomials were obtained by Luo (see [11, 12]) using Lipschitz summation, while Bayad [3] obtained Fourier expansion for the Apostol–Bernoulli, Apostol–Euler, and Apostol–Genocchi polynomials using complex analysis theory of residues. Following Luo [12] and Bayad [3], the Fourier expansion of Apostol Frobenius–Euler polynomials was derived by Araci and Acikgoz [2]. Fourier series of periodic Genocchi functions and construction of good links between Genocchi functions and zeta function were also obtained in [1]. Fourier series of higher-order Bernoulli and Euler polynomials were used by López and Temme [10] to obtain asymptotic approximations of these polynomials. Using the method in [10], approximations for higher-order Genocchi polynomials were derived in [4].
In this paper, Fourier expansions for higher-order Apostol–Genocchi, Apostol–Bernoulli, and Apostol–Euler polynomials are derived as no Fourier expansions of these polynomials are available in the literature. The method of López and Temme [10] is used to derive the desired Fourier expansions. It is found out that the method using Lipschitz summation is not applicable to these higher-order polynomials. Moreover, it is shown that for \(m=1\) the Fourier series obtained reduce to those obtained in [3] and [12] . Exceptional values of the parameter λ are also considered.
2 Fourier expansions
In this section Fourier expansions for higher-order Apostol-type polynomials mentioned above are presented and proved.
Theorem 2.1
For\(\lambda\in\mathbb{C}\), \(\lambda\neq0, -1\), \(0< z<1\), and\(n\geq m\),
where\(B_{\nu}^{m}(z)=B_{\nu}^{m}(z;1)\)denotes the Bernoulli polynomials of higher order defined in (1.2).
Proof
Applying the Cauchy integral formula to (1.1),
where C is a circle about zero with \(\text{radius} <|i\pi- \log\lambda|\). Let
Note that 0 is a pole of order \(n-m+1\), while the values \(w_{k}\) such that \(\lambda e^{w_{k}}+1=0\) are poles of order m. For \(k\in\mathbb{Z}\),
Let \(C_{k}\) be a circle about 0 with \(\mathrm{radius} <|w_{k}|\). Letting \(k\rightarrow\infty\) and using the residue theorem,
where \(R_{k}=\operatorname{Res}(f(w),w_{k})\).
For \(0< z<1\), the limit on the left-hand side of (2.5) is 0. For \(k=0\),
Then (2.5) becomes
To compute the residues \(R_{k}\), \(k\ge1\), the Laurent series of \(f(w)\) about \(w_{k}\) will be used. Since \(w_{k}\) is a pole of order m, its Laurent series is
where \(a_{-1}=\operatorname{Res}(f(w),w_{k})\).
Multiplying both sides of (2.7) by \((w-w_{k})^{m}\), we have
where \(a_{-1}\) is now the coefficient of \((w-w_{k})^{m-1}\). That is, \(a_{-1}=a_{m-1}\) in the expansion
Let
where \(\beta_{k}^{m}(n,z)\) are to be determined. From [3] and [12],
it is seen that \(\beta_{k}^{1}(n,z)=1\), ∀k.
To find an explicit formula for \(\beta_{k}^{m}(n,z)\), substitute \(w_{k}=-\log\lambda+(2k+1)\pi i\) to (2.8) and use \(f(z)\) in (2.3) to give
Let \(s=w-[-\log\lambda+(2k+1)\pi i]\). Then \(w=s-\log\lambda +(2k+1)\pi i\) and (2.11) becomes
Using (1.2) and writing
the left-hand side of (2.12) becomes
Applying Cauchy-product on (2.13) will yield
Thus,
In particular,
Substituting (2.16) to (2.17),
Using the identity
(2.18) becomes
Substituting to (2.9), the desired Fourier expansion for \(G_{n}^{m}(z;\lambda)\) is obtained. □
Remark 2.2
When \(m=1\), (2.1) reduces to
which coincides with that of Luo [12] and Bayad [3].
Theorem 2.3
For\(\lambda\in\mathbb{C}\), \(\lambda\neq0, 1\), \(0< z<1\), and\(n\geq m\),
Proof
The method used in proving Theorem 2.1 will be applied here. Applying the Cauchy integral formula to (1.2), we obtain
where C is a circle about zero with \(\text{radius}<|\log\lambda|\).
Let
Note that zero is a pole of order \(n-m+1\), while the values \(u_{k}\) such that \(\lambda e^{u_{k}}-1=0\) are poles of order m. For \(k\in\mathbb{Z}\),
Let \(C_{k}\) be a circle about 0 with \(\text{radius}<|w_{k}|\). Letting \(k\to \infty\) and using the residue theorem,
where \(S_{k}=\operatorname{Res}(g(w),u_{k})\).
For \(0< z<1\), the limit on the left-hand side of (2.25) is 0 and
Then (2.25) becomes
To compute the residues \(S_{k}\), use the Laurent series of \(g(w)\) about \(u_{k}\). Since \(u_{k}\) is a pole of order m, the Laurent series of \(g(w)\) about \(u_{k}\) is
where \(b_{-1}=\operatorname{Res}(g(w),u_{k})\).
Multiplying both sides of (2.27) by \((w-u_{k})^{m}\),
where \(b_{-1}\) is now the coefficient of \((w-u_{k})^{m-1}\). That is, \(b_{-1}=b_{m-1}\) in the expansion
Let
where \(\gamma_{k}^{m}(n,z)\) are to be determined. From [3],
it is seen that \(\gamma_{k}^{1}(n,z)=-1\), ∀k.
To find an explicit formula for \(\gamma_{k}^{m}(n,z)\), substitute \(u_{k}=-\log\lambda+ 2k \pi i\) and the function \(g(w)\) in (2.23) to (2.28) to obtain
Let \(t=w-[-\log\lambda+2k\pi i]\). Then \(w=t-\log\lambda+2k\pi i\) and (2.31) becomes
Using (1.2) and writing
the left-hand side of (2.32) becomes
Applying Cauchy-product on (2.33) will yield
Thus,
In particular,
Substituting (2.36) to (2.37),
Using the identity in (2.19), we have
Substituting (2.39) to (2.29), the desired Fourier expansion of \(B^{m}_{n}(z;\lambda)\) is obtained. □
Remark 2.4
When \(m=1\), (2.21) reduces to
which coincides with that in [3].
Theorem 2.5
For\(\lambda\in\mathbb{C}\), \(\lambda\neq0,-1\), \(0< z<1\), and\(n \geq m\),
Proof
Multiplying both sides of (1.3) by \(w^{m}\) yields
The left hand-side of (2.42) can be written
Thus,
Comparing coefficients in (2.45) gives
Using (2.1),
Simplifying
and substituting to (2.47), the desired result is obtained. □
Remark 2.6
If \(m=1\), (2.40) reduces to
which coincides with the corresponding result in [3].
3 The cases \(\lambda=-1\) and \(\lambda=1\)
Theorem 2.1 does not apply when \(\lambda=-1\) because for \(\lambda=-1\), \(w_{k}=0\), ∀k, while Theorem 2.3 does not apply for \(\lambda=1\) for similar reason. So these cases are considered here. Using (1.2),
On the other hand, using (1.1), we get
Thus,
Also, from (2.43),
We proceed to finding the Fourier expansion for \(B_{n}^{m}(z;1)\). The method in the previous section will be applied. First consider \(m=1\). The Fourier expansion for \(B_{n}^{1}(z;1)=B_{n}(z;1)\) is given in the following lemma.
Lemma 3.1
For\(0< z<1\)and\(n\geq1\),
Proof
By (1.2)
where C is a circle about the origin with \(\text{radius}<2\pi\). Let \(f(w)=\frac{e^{wz}}{(e^{w}-1)w^{n}}\). Following the method in the previous section, we obtain
where \(R_{k}=\operatorname{Res}(f(w),2k\pi i)\), \(k=\pm1,\pm2, \dots \).
These residues can be computed to be
Thus,
□
For \(m>1\), the Fourier series of \(B_{n}^{m}\)(z;1) is given in the following theorem.
Theorem 3.2
For\(0< z<1\)and\(n\geq m>1\),
Proof
By the Cauchy integral formula,
where C is a circle about the origin with \(\text{radius}<2\pi\).
The complex numbers \(u_{k}=2k\pi i\), \(k=\pm1,\pm2, \ldots \) are poles of order m of the function
Then
where \(R_{k}=\operatorname{Res}(h(w),2k\pi i)\), \(k=\pm1,\pm2,\dots \).
Let
be the Laurent series of \(h(w)\), where
Multiplying both sides of (3.9) by \((w-u_{k})^{m}\) gives
where \(c_{-1}\) is now the coefficient of \((w-u_{k})^{m-1}\).
That is, \(c_{-1}=c_{m-1}\) in the expansion
Following (3.4), write
where \(\gamma_{k}^{m}(n,z;1)\) are to be determined. Note that \(\gamma _{k}^{1}(n,z;1)=1\) (see (3.4)). From (3.11),
Let \(t=w-2k\pi i\). Then \(w=t+2k\pi i\) and (3.13) becomes
Writing
and using (3.1), (3.14) yields
Applying Cauchy-product, (3.15) becomes
Thus,
In particular,
Applying (2.19),
Substituting to (3.12), the theorem follows. □
Remark 3.3
When \(m=1\), the formula in Lemma 3.1 and Theorem 3.2 agrees with that obtained in [3].
Using (3.2) and (3.3) the following corollary is a direct consequence of Theorem 3.2.
Corollary 3.4
For\(0< z<1\)and\(n\geq m>1\),
4 Conclusion
It is seen that the Fourier expansions for higher-order Apostol–Genocchi, Apostol–Bernoulli, and Apostol–Euler polynomials are readily obtained using the method of Lopez and Temme [10]. Following [12] and [10] it will be interesting to consider the integral representations and asymptotic approximations of these polynomials for future study.
References
Araci, S., Acikgoz, M.: Applications of Fourier series and zeta functions to Genocchi polynomials. Appl. Math. Inf. Sci. 12(5), 951–955 (2018)
Araci, S., Acikgoz, M.: Construction of Fourier expansion of Apostol Frobenius–Euler polynomials and its application. Adv. Differ. Equ. 2018, Article ID 67 (2018). https://doi.org/10.1186/s13662-018-1526-x
Bayad, A.: Fourier expansions for Apostol–Bernoulli, Apostol–Euler and Apostol–Genocchi polynomials. Math. Comput. 80(276), 2219–2221 (2011). https://doi.org/10.1090/S0025-5718-2011-02476-2
Corcino, C., Corcino, R.: Asymptotics of Genocchi polynomials and higher order Genocchi polynomials using residues. Afr. Math. (2020). https://doi.org/10.1007/s13370-019-00759-z
Fixed Point (https://math.stackechange.com/users/30261/fixed-point): Real world application of Fourier series. Nov. 24, 2013. https://math.stackexchange.com/q/579695
He, Y., Araci, S., Srivastava, H.M.: Some new formulas for the products of the Apostol type polynomials. Adv. Differ. Equ. 2016, Article ID 287 (2016). https://doi.org/10.1186/s13662-016-1014-0
He, Y., Araci, S., Srivastava, H.M., Abdel-Aty, M.: Higher-order convolutions for Apostol–Bernoulli, Apostol–Euler and Apostol–Genocchi polynomials. Mathematics 6, Article ID 329 (2019). https://doi.org/10.3390/math6120329
He, Y., Araci, S., Srivastava, H.M., Acikgoz, M.: Some new identities for the Apostol–Bernoulli polynomials and the Apostol–Genocchi polynomials. Appl. Math. Comput. 262, 31–41 (2015). https://doi.org/10.1016/j.amc.2015.03.132
Hollingsworth, M.: Applications of the Fourier series (2019)
López, J.L., Temme, N.M.: Large degree asymptotics of generalized Bernoulli and Euler polynomials. J. Math. Anal. Appl. 363(1), 197–208 (2010). https://doi.org/10.1016/j.jmaa.2009.08.034
Luo, Q.-M.: Fourier expansions and integral representations for Genocchi polynomials. J. Integer Seq. 12, Article ID 09.1.4 (2009)
Luo, Q.-M.: Extensions of the Genocchi polynomials and their Fourier expansions and integral representations. Osaka J. Math. 48, 291–309 (2011)
Acknowledgements
The authors would like to thank Cebu Normal University for the financial support to this research project.
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This research project is partially funded by Cebu Normal University.
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CC was the one who conceptualized the problem and the method to be used in solving the problem. She did the introduction and derived the Fourier expansion of higher-order Apostol–Genocchi and Apostol–Bernoulli polynomials. RC derived the Fourier expansion of higher-order Apostol–Euler polynomials, and he wrote Sect. 3. All authors read and approved the final manuscript.
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Corcino, C.B., Corcino, R.B. Fourier expansions for higher-order Apostol–Genocchi, Apostol–Bernoulli and Apostol–Euler polynomials. Adv Differ Equ 2020, 346 (2020). https://doi.org/10.1186/s13662-020-02802-x
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DOI: https://doi.org/10.1186/s13662-020-02802-x