The Lerch Zeta and Related Functions of Non-Positive Integer Order
Апстракт
It is known that the Lerch (or periodic) zeta function of non-positive integer order, l(-n)(xi), n is an element of N(0) := {0, 1, 2, 3, ...}, is a polynomial in cot(pi xi) of degree n+1. In this paper, a very simple explicit closed-form formula for this polynomial valid for any degree is derived. In addition, novel analogous explicit closed-form formulae for the Legendre chi function, the alternating Lerch zeta. function and the alternating Legendre chi function are established. The obtained formulae involve the Carlitz-Scoville tangent and secant numbers of higher order, and the derivative polynomials for tangent and secant are used in their derivation. Several special cases and consequences are pointed out, and some examples arc, also given.
Кључне речи:
Lerch zeta function / Legendre chi function / alternating Lerch zeta function / alternating Legendre chi function / derivative polynomials / tangent numbers of order k / secant numbers of order k / higher (generalized) tangent numbers / higher (generalized) secant numbersИзвор:
Proceedings of the American Mathematical Society, 2010, 138, 3, 827-836Финансирање / пројекти:
- Ministry of Science of the Republic of Serbia [142025, 144004]
DOI: 10.1090/S0002-9939-09-10116-8
ISSN: 0002-9939
WoS: 000275015700006
Scopus: 2-s2.0-77951490776
Колекције
Институција/група
VinčaTY - JOUR AU - Cvijović, Đurđe PY - 2010 UR - https://vinar.vin.bg.ac.rs/handle/123456789/3919 AB - It is known that the Lerch (or periodic) zeta function of non-positive integer order, l(-n)(xi), n is an element of N(0) := {0, 1, 2, 3, ...}, is a polynomial in cot(pi xi) of degree n+1. In this paper, a very simple explicit closed-form formula for this polynomial valid for any degree is derived. In addition, novel analogous explicit closed-form formulae for the Legendre chi function, the alternating Lerch zeta. function and the alternating Legendre chi function are established. The obtained formulae involve the Carlitz-Scoville tangent and secant numbers of higher order, and the derivative polynomials for tangent and secant are used in their derivation. Several special cases and consequences are pointed out, and some examples arc, also given. T2 - Proceedings of the American Mathematical Society T1 - The Lerch Zeta and Related Functions of Non-Positive Integer Order VL - 138 IS - 3 SP - 827 EP - 836 DO - 10.1090/S0002-9939-09-10116-8 ER -
@article{ author = "Cvijović, Đurđe", year = "2010", abstract = "It is known that the Lerch (or periodic) zeta function of non-positive integer order, l(-n)(xi), n is an element of N(0) := {0, 1, 2, 3, ...}, is a polynomial in cot(pi xi) of degree n+1. In this paper, a very simple explicit closed-form formula for this polynomial valid for any degree is derived. In addition, novel analogous explicit closed-form formulae for the Legendre chi function, the alternating Lerch zeta. function and the alternating Legendre chi function are established. The obtained formulae involve the Carlitz-Scoville tangent and secant numbers of higher order, and the derivative polynomials for tangent and secant are used in their derivation. Several special cases and consequences are pointed out, and some examples arc, also given.", journal = "Proceedings of the American Mathematical Society", title = "The Lerch Zeta and Related Functions of Non-Positive Integer Order", volume = "138", number = "3", pages = "827-836", doi = "10.1090/S0002-9939-09-10116-8" }
Cvijović, Đ.. (2010). The Lerch Zeta and Related Functions of Non-Positive Integer Order. in Proceedings of the American Mathematical Society, 138(3), 827-836. https://doi.org/10.1090/S0002-9939-09-10116-8
Cvijović Đ. The Lerch Zeta and Related Functions of Non-Positive Integer Order. in Proceedings of the American Mathematical Society. 2010;138(3):827-836. doi:10.1090/S0002-9939-09-10116-8 .
Cvijović, Đurđe, "The Lerch Zeta and Related Functions of Non-Positive Integer Order" in Proceedings of the American Mathematical Society, 138, no. 3 (2010):827-836, https://doi.org/10.1090/S0002-9939-09-10116-8 . .
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