Article
| Title: | Complete monotonicity of the remainder in an asymptotic series related to the psi function (English) |
| Author: | Yang, Zhen-Hang |
| Author: | Tian, Jing-Feng |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 74 |
| Issue: | 1 |
| Year: | 2024 |
| Pages: | 337-351 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | Let $p,q\in \mathbb {R}$\ with $p-q\geq 0$, $\sigma = \frac 12 ( p+q-1)$ and $s=\frac 12 ( 1-p+q)$, and let $$ \mathcal {D}_{m} ( x;p,q ) =\mathcal {D}_{0} ( x;p,q ) +\sum _{k=1}^{m}\frac {B_{2k} ( s) }{2k ( x+\sigma ) ^{2k}} , $$ where $$ \mathcal {D}_{0} ( x;p,q ) =\frac {\psi ( x+p ) +\psi ( x+q ) }{2}-\ln ( x+\sigma ) . $$ We establish the asymptotic expansion $$ \mathcal {D}_{0} ( x;p,q ) \sim -\sum _{n=1}^{\infty } \frac {B_{2n} ( s ) }{2n ( x+\sigma ) ^{2n}} \quad \text {as} \^^Mx\rightarrow \infty , $$ where $B_{2n} ( s ) $ stands for the Bernoulli polynomials. Further, we prove that the functions $( -1) ^{m}\mathcal {D}_{m} ( x;p,q )$ and $( -1) ^{m+1}\mathcal {D}_{m} ( x;p,q )$ are completely monotonic in $x$ on $( -\sigma ,\infty )$ for every $m\in \mathbb {N}_{0}$ if and only if $p-q\in [ 0, \tfrac 12 ]$ and $p-q=1$, respectively. This not only unifies the two known results but also yields some new results. (English) |
| Keyword: | psi function |
| Keyword: | asymptotic expansion |
| Keyword: | complete monotonicity |
| MSC: | 26A48 |
| MSC: | 33B15 |
| MSC: | 41A60 |
| DOI: | 10.21136/CMJ.2024.0354-23 |
| . | |
| Date available: | 2024-03-13T10:12:37Z |
| Last updated: | 2024-03-18 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152284 |
| . | |
| Reference: | [1] Abramowitz, M., (eds.), I. A. Stegun: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.National Bureau of Standards, Applied Mathematics Series 55. John Wiley, New York (1972). Zbl 0543.33001, MR 0208798 |
| Reference: | [2] Alzer, H.: On some inequalities for the gamma and psi functions.Math. Comput. 66 (1997), 373-389. Zbl 0854.33001, MR 1388887, 10.1090/S0025-5718-97-00807-7 |
| Reference: | [3] Atanassov, R. D., Tsoukrovski, U. V.: Some properties of a class of logarithmically completely monotonic functions.C. R. Acad. Bulg. Sci. 41 (1988), 21-23 \99999MR99999 0939205 \goodbreak. Zbl 0658.26010, MR 0939205 |
| Reference: | [4] Chen, C.-P., Paris, R. B.: Inequalities, asymptotic expansions and completely monotonic functions related to the gamma function.Appl. Math. Comput. 250 (2015), 514-529. Zbl 1328.33001, MR 3285558, 10.1016/j.amc.2014.11.010 |
| Reference: | [5] Fields, J. L.: The uniform asymptotic expansion of a ratio of Gamma functions.Constructive Theory of Functions Publishing House of the Bulgarian Academy of Sciences, Sofia (1970), 171-176. Zbl 0263.33002, MR 0399527 |
| Reference: | [6] Frenzen, C. L.: Error bounds for asymptotic expansions of the ratio of two gamma functions.SIAM J. Math. Anal. 18 (1987), 890-896. Zbl 0625.41022, MR 0883576, 10.1137/0518067 |
| Reference: | [7] Luke, Y. L.: On the ratio of two gamma functions.Jñ\=an\=abha 9-10 (1980), 143-148. Zbl 0504.33001, MR 0683706 |
| Reference: | [8] Olver, F. W. J., Lozier, D. W., Boisvert, R. F., (eds.), C. W. Clark: NIST Handbook of Mathematical Functions.Cambridge University Press, Cambridge (2010). Zbl 1198.00002, MR 2723248, 10.1023/A:1022915830921 |
| Reference: | [9] Qi, F., Chen, C.-P.: A complete monotonicity property of the gamma function.J. Math. Anal. Appl. 296 (2004), 603-607. Zbl 1046.33001, MR 2075188, 10.1016/j.jmaa.2004.04.026 |
| Reference: | [10] Schilling, R. L., Song, R., Vondraček, Z.: Bernstein functions: Theory and Applications.de Gruyter Studies in Mathematics 37. Walter de Gruyter, Berlin (2010). Zbl 1197.33002, MR 2598208, 10.1515/9783110269338 |
| Reference: | [11] Tian, J.-F., Yang, Z.: Asymptotic expansions of Gurland's ratio and sharp bounds for their remainders.J. Math. Anal. Appl. 493 (2021), Article ID 124545, 19 pages. Zbl 1450.33006, MR 4144294, 10.1016/j.jmaa.2020.124545 |
| Reference: | [12] Widder, D. V.: The Laplace Transform.Princeton Mathematical Series 6. Princeton University Press, Princeton (1941). Zbl 0063.08245, MR 0005923 |
| Reference: | [13] Yang, Z.-H.: Approximations for certain hyperbolic functions by partial sums of their Taylor series and completely monotonic functions related to gamma function.J. Math. Anal. Appl. 441 (2016), 549-564. Zbl 1336.33005, MR 3491542, 10.1016/j.jmaa.2016.04.029 |
| Reference: | [14] Yang, Z.-H., Chu, Y.-M.: Jordan type inequalities for hyperbolic functions and their applications.J. Funct. Spaces 2015 (2015), Article ID 370979, 4 pages. Zbl 1323.26021, MR 3321607, 10.1155/2015/370979 |
| Reference: | [15] Yang, Z.-H., Tian, J.-F., Ha, M.-H.: A new asymptotic expansion of a ratio of two gamma functions and complete monotonicity for its remainder.Proc. Am. Math. Soc. 148 (2020), 2163-2178. Zbl 1435.41034, MR 4078101, 10.1090/proc/14917 |
| Reference: | [16] Yang, Z., Tian, J.-F.: Complete monotonicity of the remainder of the asymptotic series for the ratio of two gamma functions.J. Math. Anal. Appl. 517 (2023), Article ID 126649, 15 pages. Zbl 07595153, MR 4477953, 10.1016/j.jmaa.2022.126649 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
