### Abstract

We give an elementary calculus proof of the asymptotic formulas for the zeros of the q-sine and cosine functions which have been recently found numerically by Gosper and Suslov. Monotone convergent sequences of the lower and upper bounds for these zeros are constructed as an extension of our method. Improved asymptotics are found by a different method using the Lagrange inversion formula. Asymptotic formulas for the points of inflection of the basic sine and cosine functions are conjectured. Analytic continuation of the q-zeta function is discussed as an application. An interpretation of the zeros is given.

Original language | English (US) |
---|---|

Pages (from-to) | 292-335 |

Number of pages | 44 |

Journal | Journal of Approximation Theory |

Volume | 121 |

Issue number | 2 |

DOIs | |

State | Published - Apr 1 2003 |

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### Keywords

- Asymptotics of zeros of q-trigonometric functions
- Basic Fourier series
- Basic trigonometric functions
- Lagrange inversion formula
- q-zeta function

### ASJC Scopus subject areas

- Mathematics(all)
- Analysis
- Applied Mathematics
- Numerical Analysis

### Cite this

**Asymptotics of zeros of basic sine and cosine functions.** / Suslov, Sergei.

Research output: Contribution to journal › Article

*Journal of Approximation Theory*, vol. 121, no. 2, pp. 292-335. https://doi.org/10.1016/S0021-9045(03)00027-3

}

TY - JOUR

T1 - Asymptotics of zeros of basic sine and cosine functions

AU - Suslov, Sergei

PY - 2003/4/1

Y1 - 2003/4/1

N2 - We give an elementary calculus proof of the asymptotic formulas for the zeros of the q-sine and cosine functions which have been recently found numerically by Gosper and Suslov. Monotone convergent sequences of the lower and upper bounds for these zeros are constructed as an extension of our method. Improved asymptotics are found by a different method using the Lagrange inversion formula. Asymptotic formulas for the points of inflection of the basic sine and cosine functions are conjectured. Analytic continuation of the q-zeta function is discussed as an application. An interpretation of the zeros is given.

AB - We give an elementary calculus proof of the asymptotic formulas for the zeros of the q-sine and cosine functions which have been recently found numerically by Gosper and Suslov. Monotone convergent sequences of the lower and upper bounds for these zeros are constructed as an extension of our method. Improved asymptotics are found by a different method using the Lagrange inversion formula. Asymptotic formulas for the points of inflection of the basic sine and cosine functions are conjectured. Analytic continuation of the q-zeta function is discussed as an application. An interpretation of the zeros is given.

KW - Asymptotics of zeros of q-trigonometric functions

KW - Basic Fourier series

KW - Basic trigonometric functions

KW - Lagrange inversion formula

KW - q-zeta function

UR - http://www.scopus.com/inward/record.url?scp=0038706201&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0038706201&partnerID=8YFLogxK

U2 - 10.1016/S0021-9045(03)00027-3

DO - 10.1016/S0021-9045(03)00027-3

M3 - Article

AN - SCOPUS:0038706201

VL - 121

SP - 292

EP - 335

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

SN - 0021-9045

IS - 2

ER -