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840 results on '"Digamma function"'

151. A New Class of Symmetric Beta Type Distributions Constructed by Means of Symmetric Bernstein Type Basis Functions

Author

Yilmaz Simsek and Fusun Yalcin

Subjects
Pure mathematics, Physics and Astronomy (miscellaneous), symmetric beta distributions, General Mathematics, Bernstein basis functions, Basis function, 01 natural sciences, Lah number, Stirling numbers, Computer Science (miscellaneous), Stirling number, Harmonic number, digamma function, 0101 mathematics, Mathematics, Series (mathematics), lcsh:Mathematics, 010102 general mathematics, moment generating function, Moment-generating function, lcsh:QA1-939, 010101 applied mathematics, Digamma function, Chemistry (miscellaneous), Apostol–Bernoulli polynomials, symmetry property random variable, harmonic numbers, Lah numbers, Polygamma function
Abstract

The main aim of this paper is to define and investigate a new class of symmetric beta type distributions with the help of the symmetric Bernstein-type basis functions. We give symmetry property of these distributions and the Bernstein-type basis functions. Using the Bernstein-type basis functions and binomial series, we give some series and integral representations including moment generating function for these distributions. Using generating functions and their functional equations, we also give many new identities related to the moments, the polygamma function, the digamma function, the harmonic numbers, the Stirling numbers, generalized harmonic numbers, the Lah numbers, the Bernstein-type basis functions, the array polynomials, and the Apostol&ndash, Bernoulli polynomials. Moreover, some numerical values of the expected values for the logarithm of random variable are given.

Published
2020
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152. Completely monotonic ratios of basic and ordinary gamma functions

Author

Dmitrii Karp, Asena Çetinkaya, and Christian Berg

Subjects
Completely monotonic function, Pure mathematics, Logarithm, General Mathematics, Monotonic function, 010103 numerical & computational mathematics, Rational function, Logarithmic complete monotonicity, 01 natural sciences, Measure (mathematics), q-Gamma function, Simple (abstract algebra), Classical Analysis and ODEs (math.CA), FOS: Mathematics, Discrete Mathematics and Combinatorics, Elementary function, 0101 mathematics, Sherman’s theorem, Gamma function, Bernstein function, Mathematics, Variable (mathematics), 33B15, 33D05, 26A48, Applied Mathematics, 010102 general mathematics, Mathematics - Classical Analysis and ODEs, Digamma function
Abstract

We investigate conditions for logarithmic complete monotonicity of product ratios of gamma and q-gamma functions whose arguments are linear functions of the variable. We give necessary and sufficient conditions in terms of nonnegativity of a certain explicitly written measure in the q case and of a certain elementary function in the classical q=1 case. In the latter case we further provide simple new sufficient conditions leading to many new examples of logarithmically completely monotonic gamma ratios. Finally, we apply some of our results to study monotonicity of some gamma ratios and rational functions., 23 pages; no figures

Published
2020

153. Differentiation of the Mittag-Leffler Functions with Respect to Parameters in the Laplace Transform Approach

Author

Alexander Apelblat

Subjects
Power series, Work (thermodynamics), Pure mathematics, General Mathematics, Mathematics::Classical Analysis and ODEs, integral representations, 01 natural sciences, infinite power series, 010305 fluids & plasmas, Convolution, Laplace transform approach, Mathematics::Probability, 0103 physical sciences, Computer Science (miscellaneous), derivatives with respect to parameters, 010306 general physics, Gamma function, Engineering (miscellaneous), Quotient, Mathematics, Laplace transform, Series (mathematics), Mathematics::Complex Variables, lcsh:Mathematics, lcsh:QA1-939, convolution integrals, Digamma function, quotients of digamma and gamma functions, Mittag-Leffler functions
Abstract

In this work, properties of one- or two-parameter Mittag-Leffler functions are derived using the Laplace transform approach. It is demonstrated that manipulations with the pair direct&ndash, inverse transform makes it far more easy than previous methods to derive known and new properties of the Mittag-Leffler functions. Moreover, it is shown that sums of infinite series of the Mittag-Leffler functions can be expressed as convolution integrals, while the derivatives of the Mittag-Leffler functions with respect to their parameters are expressible as double convolution integrals. The derivatives can also be obtained from integral representations of the Mittag-Leffler functions. On the other hand, direct differentiation of the Mittag-Leffler functions with respect to parameters produces an infinite power series, whose coefficients are quotients of the digamma and gamma functions. Closed forms of these series can be derived when the parameters are set to be integers.

Published
2020

154. COMPLETE MONOTONICITY FOR A NEW RATIO OF FINITE MANY GAMMA FUNCTIONS

Author

Qi, Feng, Henan Polytechnic University, Tianjin Polytechnic University, and Inner Mongolia University for Nationalities

Subjects
ratio, trigamma function, inequality, complete monotonicity, linear combination, [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA], [MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT], [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], logarithmic derivative, gamma function, generating function, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], Mathematics::Metric Geometry, MSC 2010: Primary 33B15, Secondary 26A48, 26A51, 26D15, 44A10, digamma function, Bernoulli number
Abstract

In the paper, by deriving an inequality involving the generating function of the Bernoulli numbers, the author introduces a new ratio of finite many gamma functions, finds complete monotonicity of the second logarithmic derivative of the ratio, and simply reviews complete monotonicity of several linear combinations of finite many digamma or trigamma functions.

Published
2020

155. On certain properties and expansions of zeta matrix function, digamma matrix function and polygamma matrix function

Author

Ravi Dwivedi and Vivek Sahai

Subjects
Mathematics (miscellaneous), Digamma function, Matrix function, 010102 general mathematics, Convergence (routing), Mathematics::Classical Analysis and ODEs, Applied mathematics, 010103 numerical & computational mathematics, 0101 mathematics, Generalized zeta function, polygamma function, matrix functional calculus, 01 natural sciences, Polygamma function, Mathematics
Abstract

We introduce the generalized zeta matrix function, digamma matrix function and polygamma matrix function. We obtain the regions of convergence of these matrix functions, some recurrence relations and their integral representations. An identity involving hypergeometric matrix function 3F2 in the form of digamma matrix function is also given.Mathematics Subject Classification (2010): 15A15, 33B15, 11M06.Keywords: Generalized zeta function, polygamma function, matrix functional calculus

Published
2020

156. Proof of Sarkar-Kumar's Conjectures on Average Entanglement Entropies over the Bures-Hall Ensemble

Author

Lu Wei

Subjects
FOS: Computer and information sciences, Statistics and Probability, Bipartite system, Computer Science - Information Theory, Dimension (graph theory), Mathematics::Classical Analysis and ODEs, FOS: Physical sciences, General Physics and Astronomy, Quantum entanglement, Von Neumann entropy, 01 natural sciences, Measure (mathematics), 010305 fluids & plasmas, Combinatorics, 0103 physical sciences, Connection (algebraic framework), 010306 general physics, Mathematical Physics, Physics, Quantum Physics, Information Theory (cs.IT), Statistical and Nonlinear Physics, Mathematical Physics (math-ph), Digamma function, Modeling and Simulation, Quantum Physics (quant-ph), Random matrix
Abstract

Sarkar and Kumar recently conjectured [J. Phys. A: Math. Theor. $\textbf{52}$, 295203 (2019)] that for a bipartite system of Hilbert dimension $mn$, the mean values of quantum purity and von Neumann entropy of a subsystem of dimension $m\leq n$ over the Bures-Hall measure are given by \begin{equation*} \frac{2n(2n+m)-m^{2}+1}{2n(2mn-m^2+2)} \end{equation*} and \begin{equation*} \psi_{0}\left(mn-\frac{m^2}{2}+1\right)-\psi_{0}\left(n+\frac{1}{2}\right), \end{equation*} respectively, where $\psi_{0}(\cdot)$ is the digamma function. We prove the above conjectured formulas in this work. A key ingredient of the proofs is Forrester and Kieburg's discovery on the connection between the Bures-Hall ensemble and the Cauchy-Laguerre biorthogonal ensemble studied by Bertola, Gekhtman, and Szmigielski.

Published
2020

157. Measurement of the integrated luminosity of the Phase 2 data of the Belle II experiment

Author

Abudinén 40, F., Adachi 19, I., Ahlburg 100, P., Aihara 117, H., Akopov 123, N., Aloisio 89, A., Ameli 37, F., Andricek 57, L., Anh Ky 30, N., Asner 2, D. M., Atmacan 102, H., Aushev 59, T., Aushev 80, V., Aziz 81, T., Azmi 108, K., Babu 8, V., Baehr 44, S., Bahinipati 22, S., Bakich 116, A. M., Bambade 50, P., Banerjee 107, Sw., Bansal 72, S., Bansal 71, V., Barrett 19, M., Baudot 98, J., Beaulieu 119, A., Becker 44, J., Behera 24, P. K., Bennett 111, J. V., Bernieri 38, E., Bernlochner 44, F. U., Bertemes 27, M., Bessner 104, M., Bettarini 93, S., Bhardwaj 21, V., Bianchi 95, F., Bilka 5, T., Bilokin 98, S., Biswas 107, D., Bonvicini 121, G., Bozek 65, A., Bračko 109, M., Branchini 38, P., Braun 44, N., Browder 104, T. E., Budano 38, A., Bussino 94, S., Campajola 89, M., Cao 44, L., Casarosa 93, G., Cecchi 92, C., Červenkov 5, D., Chang 12, M.-C., Chang 64, P., Cheaib 101, R., Chekelian 56, V., Chen 113, Y. Q., Chen 64, Y.-T., Cheon 18, B. G., Chilikin 51, K., Cho 18, H.-E., Cho 46, K., Cho 124, S., Choi 17, S.-K., Choudhury 23, S., Cinabro 121, D., Corona 93, L., Cremaldi 111, L. M., Cunliffe 8, S., Czank 118, T., Dattola 8, F., De La Cruz-Burelo 4, E., De Nardo 89, G., De Nuccio 8, M., De Pietro 94, G., De Sangro 32, R., Destefanis 95, M., Dey 83, S., De Yta-Hernandez 4, A., Di Capua 89, F., Di Carlo 50, S., Dingfelder 100, J., Doležal 5, Z., Domínguez Jiménez 88, I., Dong 13, T. V., Dort 43, K., Dubey 104, S., Duell 100, S., Eidelman 3, S., Eliachevitch 44, M., Ferber 8, T., Ferlewicz 110, D., Finocchiaro 32, G., Fiore 37, S., Fodor 58, A., Forti 93, F., Frey 14, A., Fulsom 71, B. G., Gabriel 56, M., Ganiev 96, E., Garcia-Hernandez 4, M., Garg 72, R., Garmash 3, A., Gaur 120, V., Gaz 62, A., Gebauer 14, U., Gellrich 8, A., Gemmler 44, J., Geßler 43, T., Giordano 89, R., Giri 23, A., Gobbo 40, B., Godang 114, R., Goldenzweig 44, P., Golob 106, B., Gomis 31, P., Grace 99, P., Gradl 42, W., Graziani 38, E., Greenwald 82, D., Hadjivasiliou 71, C., Halder 81, S., Hara 19, K., Hara 19, T., Hartbrich 104, O., Hayasaka 66, K., Hayashii 63, H., Hearty 101, C., Hedges 104, M. T., Heredia De La Cruz 4, I., 7, Hernández Villanueva 111, M., Hershenhorn 101, A., Higuchi 118, T., Hill 101, E. C., Hirata 61, H., Hoek 42, M., Hollitt 99, S., Hotta 70, T., Hsu 116, C.-L., Hu 28, Y., Huang 64, K., Iijima 61, T., Inami 61, K., Inguglia 27, G., Irakkathil Jabbar 44, J., Ishikawa 19, A., Itoh 19, R., Iwasaki 69, M., Iwasaki 19, Y., Iwata 87, S., Jackson 99, P., Jacobs 25, W. W., Jaffe 2, D. E., Jang 17, E.-J., Jeon 49, H. B., Jia 1, S., Jin 40, Y., Joo 118, C., Kahn 44, J., Kakuno 87, H., Kaliyar 24, A. B., Karyan 123, G., Kato 62, Y., Kawasaki 45, T., Kichimi 19, H., Kiesling 56, C., Kim 76, B. H., Kim 18, C.-H., Kim 78, D. Y., Kim 18, S.-H., Kim 124, Y. K., Kim 47, Y., Kimmel 120, T. D., Kinoshita 102, K., Kleinwort 8, C., Knysh 50, B., Kodyš 5, P., Koga 19, T., Komarov 8, I., Konno 45, T., Korpar 109, S., Kotchetkov 104, D., Kovalchuk 8, N., Kraetzschmar 56, T. M. G., Križan 106, P., Kroeger 111, R., Krohn 110, J. F., Krokovny 3, P., Kuehn 43, W., Kuhr 53, T., Kumar 55, M., Kumar 74, R., Kumara 121, K., Kurz 8, S., Kuzmin 3, A., Kwon 124, Y.-J., Lacaprara 34, S., Lai 19, Y.-T., La Licata 118, C., Lalwani 55, K., Lanceri 40, L., Lange 43, J. S., Lautenbach 43, K., Lee 18, I.-S., Lee 49, S. C., Leitl 56, P., Levit 82, D., Lewis 100, P. M., Li 52, C., Li 28, L. K., Li 1, S. X., Li 28, Y. M., Li 73, Y. B., Libby 24, J., Lieret 53, K., Li Gioi 56, L., Lin 64, J., Liptak 104, Z., Liu 13, Q. Y., Liventsev 120, D., Longo 119, S., Loos 115, A., Luetticke 100, F., Luo 13, T., MacQueen 110, C., Maeda 62, Y., Maggiora 95, M., Maity 22, S., Manoni 35, E., Marcello 95, S., Marinas 31, C., Martini 94, A., Masuda 10, M., Matsuoka 62, K., Matvienko 3, D., McNeil 103, J., Mei 13, J. C., Meier 116, F., Merola 90, M., Metzner 44, F., Milesi 110, M., Miller 119, C., Miyabayashi 63, K., Miyata 66, H., Mizuk 51, R., Mohanty 81, G. B., Moon 47, H., Morii 118, T., Moser 56, H.-G., Mueller 56, F., Müller 8, F. J., Muller 44, Th., Mussa 39, R., Nakamura 19, K. R., Nakano 69, E., Nakao 19, M., Nakayama 19, H., Nakazawa 64, H., Nayak 83, M., Nazaryan 123, G., Neverov 61, D., Niiyama 48, M., Nisar 112, N. K., Nishida 19, S., Nishimura 104, K., Nishimura 19, M., Nouxman 108, M. H. A., Oberhof 32, B., Ogawa 84, S., Onishchuk 80, Y., Ono 66, H., Onuki 117, Y., Oskin 51, P., Ozaki 19, H., Pakhlov 51, P., Pakhlova 59, G., Paladino 93, A., Pang 112, T., Paoloni 93, E., Park 49, H., Park 124, S.-H., Paschen 100, B., Passeri 38, A., Patra 21, S., Paul 82, S., Pedlar 54, T. K., Peruzzi 32, I., Peschke 104, R., Pestotnik 79, R., Piccolo 32, M., Piilonen 120, L. E., Podesta-Lerma 88, P. L. M., Popov 59, V., Praz 8, C., Prencipe 11, E., Prim 44, M. T., Purohit 68, M. V., Rados 8, P., Remnev 3, M., Resmi 24, P. K., Ripp-Baudot 98, I., Ritter 53, M., Ritzert 105, M., Rizzo 93, G., Rizzuto 79, L. B., Robertson 58, S. H., Rodríguez Pérez 88, D., Roney 119, J. M., Rosenfeld 115, C., Rostomyan 8, A., Rout 24, N., Russo 89, G., Sahoo 81, D., Sakai 19, Y., Sanders 111, D. A., Sandilya 102, S., Sangal 102, A., Santelj 106, L., Sartori 91, P., Sato 85, Y., Savinov 112, V., Scavino 42, B., Schram 71, M., Schreeck 14, H., Schueler 104, J., Schwanda 27, C., Schwartz 102, A. J., Schwenker 14, B., Seddon 58, R. M., Seino 66, Y., Selce 35, A., Senyo 122, K., Sevior 110, M. E., Sfienti 42, C., Shen 13, C. P., Shibuya 84, H., Shiu 64, J.-G., Sibidanov 119, A., Simon 56, F., Skambraks 56, S., Sobie 119, R. J., Soffer 83, A., Sokolov 26, A., Solovieva 51, E., Spataro 95, S., Spruck 42, B., Starič 79, M., Stefkova 8, S., Stottler 120, Z. S., Stroili 91, R., Strube 71, J., Sumihama 15, M., Sumiyoshi 87, T., Summers 111, D. J., Sutcliffe 44, W., Tabata 6, M., Takizawa 77, M., Tamponi 39, U., Tanaka 19, S., Tanida 41, K., Tanigawa 117, H., Taniguchi 19, N., Tao 103, Y., Taras 97, P., Tenchini 8, F., Torassa 34, E., Trabelsi 50, K., Tsuboyama 19, T., Tsuzuki 61, N., Uchida 86, M., Ueda 19, I., Uehara 19, S., Uglov 51, T., Unger 44, K., Unno 18, Y., Uno 19, S., Urquijo 110, P., Ushiroda 19, Y., 117, Vahsen 104, S. E., Van Tonder 44, R., Varner 104, G. S., Varvell 116, K. E., Vinokurova 3, A., Vitale 96, L., Vossen 9, A., Waheed 110, E., Wakeling 58, H. M., Wan 117, K., Wan Abdullah 108, W., Wang 56, B., Wang 64, M.-Z., Wang 13, X. L., Warburton 58, A., Watanabe 66, M., Watanuki 50, S., Webb 110, J., Wehle 8, S., Wermes 100, N., Wessel 100, C., Wiechczynski 36, J., Wieduwilt 14, P., Windel 56, H., Won 47, E., Yamada 19, S., Yan 113, W., Yang 47, S. B., Ye 8, H., Yelton 103, J., Yin 28, J. H., Yonenaga 87, M., Yook 28, Y. M., Yuan 28, C. Z., Yusa 66, Y., Zani 93, L., Zhang 28, J. Z., Zhang 113, Z., Zhilich 3, V., Zhou 19, Q. D., Zhou 1, X. Y., Zhukova 51, V. I., Zhulanov 3, V., Zupanc 109, A., Laboratoire de l'Accélérateur Linéaire (LAL), Université Paris-Sud - Paris 11 (UP11)-Institut National de Physique Nucléaire et de Physique des Particules du CNRS (IN2P3)-Centre National de la Recherche Scientifique (CNRS), Institut Pluridisciplinaire Hubert Curien (IPHC), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Belle-II, Abudinén, F., Adachi, I., Ahlburg, P., Aihara, H., Akopov, N., Aloisio, A., Ameli, F., Andricek, L., Anh Ky, N., Asner, D. M., Atmacan, H., Aushev, T., Aushev, V., Aziz, T., Azmi, K., Babu, V., Baehr, S., Bahinipati, S., Bakich, A. M., Bambade, P., Banerjee, Sw., Bansal, S., Bansal, V., Barrett, M., Baudot, J., Beaulieu, A., Becker, J., Behera, P. K., Bennett, J. V., Bernieri, E., Bernlochner, F. U., Bertemes, M., Bessner, M., Bettarini, S., Bhardwaj, V., Bianchi, F., Bilka, T., Bilokin, S., Biswas, D., Bonvicini, G., Bozek, A., Bračko, M., Branchini, P., Braun, N., Browder, T. E., Budano, A., Bussino, S., Campajola, M., Cao, L., Casarosa, G., Cecchi, C., Červenkov, D., Chang, M. -C., Chang, P., Cheaib, R., Chekelian, V., Chen, Y. Q., Chen, Y. -T., Cheon, B. G., Chilikin, K., Cho, H. -E., Cho, K., Cho, S., Choi, S. -K., Choudhury, S., Cinabro, D., Corona, L., Cremaldi, L. M., Cunliffe, S., Czank, T., Dattola, F., De La Cruz-Burelo, E., De Nardo, G., De Nuccio, M., De Pietro, G., de Sangro, R., Destefanis, M., Dey, S., De Yta-Hernandez, A., Di Capua, F., Di Carlo, S., Dingfelder, J., Doležal, Z., Domínguez Jiménez, I., Dong, T. V., Dort, K., Dubey, S., Duell, S., Eidelman, S., Eliachevitch, M., Ferber, T., Ferlewicz, D., Finocchiaro, G., Fiore, S., Fodor, A., Forti, F., Frey, A., Fulsom, B. G., Gabriel, M., Ganiev, E., Garcia-Hernandez, M., Garg, R., Garmash, A., Gaur, V., Gaz, A., Gebauer, U., Gellrich, A., Gemmler, J., Geßler, T., Giordano, R., Giri, A., Gobbo, B., Godang, R., Goldenzweig, P., Golob, B., Gomis, P., Grace, P., Gradl, W., Graziani, E., Greenwald, D., Hadjivasiliou, C., Halder, S., Hara, K., Hara, T., Hartbrich, O., Hayasaka, K., Hayashii, H., Hearty, C., Hedges, M. T., Heredia de la Cruz, I., Hernández Villanueva, M., Hershenhorn, A., Higuchi, T., Hill, E. C., Hirata, H., Hoek, M., Hollitt, S., Hotta, T., Hsu, C. -L., Hu, Y., Huang, K., Iijima, T., Inami, K., Inguglia, G., Irakkathil Jabbar, J., Ishikawa, A., Itoh, R., Iwasaki, M., Iwasaki, Y., Iwata, S., Jackson, P., Jacobs, W. W., Jaffe, D. E., Jang, E. -J., Jeon, H. B., Jia, S., Jin, Y., Joo, C., Kahn, J., Kakuno, H., Kaliyar, A. B., Karyan, G., Kato, Y., Kawasaki, T., Kichimi, H., Kiesling, C., Kim, B. H., Kim, C. -H., Kim, D. Y., Kim, S. -H., Kim, Y. K., Kim, Y., Kimmel, T. D., Kinoshita, K., Kleinwort, C., Knysh, B., Kodyš, P., Koga, T., Komarov, I., Konno, T., Korpar, S., Kotchetkov, D., Kovalchuk, N., Kraetzschmar, T. M. G., Križan, P., Kroeger, R., Krohn, J. F., Krokovny, P., Kuehn, W., Kuhr, T., Kumar, M., Kumar, R., Kumara, K., Kurz, S., Kuzmin, A., Kwon, Y. -J., Lacaprara, S., Lai, Y. -T., La Licata, C., Lalwani, K., Lanceri, L., Lange, J. S., Lautenbach, K., Lee, I. -S., Lee, S. C., Leitl, P., Levit, D., Lewis, P. M., Li, C., Li, L. K., Li, S. X., Li, Y. M., Li, Y. B., Libby, J., Lieret, K., Li Gioi, L., Lin, J., Liptak, Z., Liu, Q. Y., Liventsev, D., Longo, S., Loos, A., Luetticke, F., Luo, T., Macqueen, C., Maeda, Y., Maggiora, M., Maity, S., Manoni, E., Marcello, S., Marinas, C., Martini, A., Masuda, M., Matsuoka, K., Matvienko, D., Mcneil, J., Mei, J. C., Meier, F., Merola, M., Metzner, F., Milesi, M., Miller, C., Miyabayashi, K., Miyata, H., Mizuk, R., Mohanty, G. B., Moon, H., Morii, T., Moser, H. -G., Mueller, F., Müller, F. J., Muller, Th., Mussa, R., Nakamura, K. R., Nakano, E., Nakao, M., Nakayama, H., Nakazawa, H., Nayak, M., Nazaryan, G., Neverov, D., Niiyama, M., Nisar, N. K., Nishida, S., Nishimura, K., Nishimura, M., Nouxman, M. H. A., Oberhof, B., Ogawa, S., Onishchuk, Y., Ono, H., Onuki, Y., Oskin, P., Ozaki, H., Pakhlov, P., Pakhlova, G., Paladino, A., Pang, T., Paoloni, E., Park, H., Park, S. -H., Paschen, B., Passeri, A., Patra, S., Paul, S., Pedlar, T. K., Peruzzi, I., Peschke, R., Pestotnik, R., Piccolo, M., Piilonen, L. E., Podesta-Lerma, P. L. M., Popov, V., Praz, C., Prencipe, E., Prim, M. T., Purohit, M. V., Rados, P., Remnev, M., Resmi, P. K., Ripp-Baudot, I., Ritter, M., Ritzert, M., Rizzo, G., Rizzuto, L. B., Robertson, S. H., Rodríguez Pérez, D., Roney, J. M., Rosenfeld, C., Rostomyan, A., Rout, N., Russo, G., Sahoo, D., Sakai, Y., Sanders, D. A., Sandilya, S., Sangal, A., Santelj, L., Sartori, P., Sato, Y., Savinov, V., Scavino, B., Schram, M., Schreeck, H., Schueler, J., Schwanda, C., Schwartz, A. J., Schwenker, B., Seddon, R. M., Seino, Y., Selce, A., Senyo, K., Sevior, M. E., Sfienti, C., Shen, C. P., Shibuya, H., Shiu, J. -G., Sibidanov, A., Simon, F., Skambraks, S., Sobie, R. J., Soffer, A., Sokolov, A., Solovieva, E., Spataro, S., Spruck, B., Starič, M., Stefkova, S., Stottler, Z. S., Stroili, R., Strube, J., Sumihama, M., Sumiyoshi, T., Summers, D. J., Sutcliffe, W., Tabata, M., Takizawa, M., Tamponi, U., Tanaka, S., Tanida, K., Tanigawa, H., Taniguchi, N., Tao, Y., Taras, P., Tenchini, F., Torassa, E., Trabelsi, K., Tsuboyama, T., Tsuzuki, N., Uchida, M., Ueda, I., Uehara, S., Uglov, T., Unger, K., Unno, Y., Uno, S., Urquijo, P., Ushiroda, Y., Vahsen, S. E., van Tonder, R., Varner, G. S., Varvell, K. E., Vinokurova, A., Vitale, L., Vossen, A., Waheed, E., Wakeling, H. M., Wan, K., Wan Abdullah, W., Wang, B., Wang, M. -Z., Wang, X. L., Warburton, A., Watanabe, M., Watanuki, S., Webb, J., Wehle, S., Wermes, N., Wessel, C., Wiechczynski, J., Wieduwilt, P., Windel, H., Won, E., Yamada, S., Yan, W., Yang, S. B., Ye, H., Yelton, J., Yin, J. H., Yonenaga, M., Yook, Y. M., Yuan, C. Z., Yusa, Y., Zani, L., Zhang, J. Z., Zhang, Z., Zhilich, V., Zhou, Q. D., Zhou, X. Y., Zhukova, V. I., Zhulanov, V., Zupanc, A., Centre National de la Recherche Scientifique (CNRS)-Institut National de Physique Nucléaire et de Physique des Particules du CNRS (IN2P3)-Université Paris-Sud - Paris 11 (UP11), Université de Strasbourg (UNISTRA)-Institut National de Physique Nucléaire et de Physique des Particules du CNRS (IN2P3)-Centre National de la Recherche Scientifique (CNRS), Belle II, Collaboration, Russo, Guido, Et, Al., Ministerio de Ciencia, Innovación y Universidades (España), Generalitat Valenciana, and II Collaboration, Belle

Subjects
BELLE: upgrade, Nuclear and High Energy Physics, Particle physics, data analysis method, Astrophysics::High Energy Astrophysical Phenomena, Phase (waves), FOS: Physical sciences, Computer Science::Computational Geometry, elastic scattering [electron positron], 01 natural sciences, Resonance (particle physics), High Energy Physics - Experiment, law.invention, High Energy Physics - Experiment (hep-ex), KEK-B, law, electron positron: elastic scattering, 0103 physical sciences, [PHYS.HEXP]Physics [physics]/High Energy Physics - Experiment [hep-ex], Bhabha, Belle II, ddc:530, luminosity, 010306 general physics, Collider, Instrumentation, Monte Carlo, digamma, upgrade [BELLE], Physics, Measure (data warehouse), Luminosity (scattering theory), 010308 nuclear & particles physics, Detector, Astronomy and Astrophysics, two-photon [final state], ddc, Digamma function, High Energy Physics::Experiment, final state: two-photon, Energy (signal processing), experimental results
Abstract

Chinese physics / C 44(2), 021001 (2020). doi:10.1088/1674-1137/44/2/021001, From April to July 2018, a data sample at the peak energy of the Υ(4S) resonance was collected with the Belle II detector at the SuperKEKB electron-positron collider. This is the first data sample of the Belle II experiment. Using Bhabha and digamma events, we measure the integrated luminosity of the data sample to be (496.3±0.3±3.0)pb−1, where the first uncertainty is statistical and the second is systematic. This work provides a basis for future luminosity measurements at Belle II.Key words: luminosity , Bhabha , digamma , Belle II, Published by IOP Publ., Bristol [u.a.]

Published
2020
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158. Is F-shaped digamma attested as a numerical sign in Greek papyri? Once more on P.Herc. 1669 and P.Oxy. 1176

Author

Graziano Ranocchia

Subjects
Linguistics and Language, Archeology, Literature and Literary Theory, Visual Arts and Performing Arts, Milesian numeral system, media_common.quotation_subject, Digamma, P.Oxy. 1176, Language and Linguistics, Sign (linguistics), Numeral system, Functional evolution, Reading (process), P.Herc. 1669, Stigma, Classics, Satyrus, media_common, Literature, biology, business.industry, Philosophy, biology.organism_classification, Symbol, Digamma function, Rhetoric, business
Abstract

T. Dorandi, who has previously proposed to read the book number included in the end-title of P.Herc. 1669 (Philodemus’ On Rhetoric) as a numerical ϝ-shaped digamma (= 6), has now advanced the same reading in the subscriptio of P.Oxy. 1176 (Satyrus, Lives book 6), where the editor princeps and all subsequent editors had unanimously read a stigma before. In this article, I argue not only that both readings are palaeographically untenable, but also that they historically contradict the graphic and functional evolution of digamma within the Greek alphabet. In particular, in both Graeco-Egyptian and Herculaneum papyri, ϝ-shaped digamma is always attested as a merely phonetic element (/w/ or waw) – never as a numeral symbol – whereas stigma, which represents the historical evolution of a variant of Archaic and Classical digamma, is the only form used as a numerical sign (= 6).

Published
2020

159. Degenerate Analogues of Euler Zeta, Digamma, and Polygamma Functions

Author

Fuli He, Ahmed Bakhet, Mohamed Abdalla, and Mohamed Akel

Subjects
Pure mathematics, Recurrence relation, Article Subject, General Mathematics, 010102 general mathematics, Degenerate energy levels, General Engineering, Mathematics::Classical Analysis and ODEs, 010103 numerical & computational mathematics, Engineering (General). Civil engineering (General), 01 natural sciences, Riemann zeta function, symbols.namesake, Digamma function, Special functions, QA1-939, Euler's formula, symbols, TA1-2040, 0101 mathematics, Hypergeometric function, Polygamma function, Mathematics
Abstract

In recent years, much attention has been paid to the role of degenerate versions of special functions and polynomials in mathematical physics and engineering. In the present paper, we introduce a degenerate Euler zeta function, a degenerate digamma function, and a degenerate polygamma function. We present several properties, recurrence relations, infinite series, and integral representations for these functions. Furthermore, we establish identities involving hypergeometric functions in terms of degenerate digamma function.

Published
2020
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160. General infinite series evaluations involving Fibonacci numbers and the Riemann Zeta function

Author

Robert Frontczak and Taras Goy

Subjects
Physics, Fibonacci number, Mathematics - Number Theory, Series (mathematics), Mathematics::General Mathematics, General Mathematics, Mathematics::Number Theory, 010102 general mathematics, Mathematics::History and Overview, Generating function, 11B39, 11M06, 40C15, 01 natural sciences, Riemann zeta function, 010101 applied mathematics, Combinatorics, symbols.namesake, Digamma function, Integer, Lucas number, symbols, FOS: Mathematics, Number Theory (math.NT), 0101 mathematics, Constant (mathematics)
Abstract

The purpose of this article is to present closed forms for various types of infinite series involving Fibonacci (Lucas) numbers and the Riemann zeta function at integer arguments., Comment: 9 pages

Published
2020
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161. Some properties of a hypergeometric function which appear in an approximation problem.

Author

Milovanović, Gradimir and Rassias, Michael

Subjects
APPROXIMATION error, MATHEMATICAL functions, HYPERGEOMETRIC functions, GAUSSIAN quadrature formulas, POLYNOMIAL approximation, EXPONENTIAL functions
Abstract

In this paper we consider properties and power expressions of the functions $$f:(-1,1)\rightarrow \mathbb{R }$$ and $$f_L:(-1,1)\rightarrow \mathbb{R }$$, defined by respectively, where $$\gamma $$ is a real parameter, as well as some properties of a two parametric real-valued function $$D(\,\cdot \,;\alpha ,\beta ) :(-1,1) \rightarrow \mathbb{R }$$, defined by The inequality of Turán type for $$\alpha +\beta >0$$ is proved, as well as an opposite inequality if $$\alpha +\beta <0$$. Finally, for the partial derivatives of $$D(x;\alpha ,\beta )$$ with respect to $$\alpha $$ or $$\beta $$, respectively $$A(x;\alpha ,\beta )$$ and $$B(x;\alpha ,\beta )$$, for which $$A(x;\alpha ,\beta )=B(x;-\beta ,-\alpha )$$, some results are obtained. We mention also that some results of this paper have been successfully applied in various problems in the theory of polynomial approximation and some 'truncated' quadrature formulas of Gaussian type with an exponential weight on the real semiaxis, especially in a computation of Mhaskar-Rahmanov-Saff numbers. [ABSTRACT FROM AUTHOR]

Published
2013
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162. Completely monotonic degrees of remainders of asymptotic expansions of the digamma function

Author

Qi, Feng, Mahmoud, Mansour, Henan Polytechnic University, Tianjin Polytechnic University, Inner Mongolia University for Nationalities, and Mansoura University [Egypt]

Subjects
remainder, completely monotonic degree, asymptotic expansion, Qi's conjecture, completely monotonic function, digamma function, [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA], logarithm of the gamma function, MSC: Primary 33B15, Secondary 26A48, 41A60, 44A10
Abstract

Motivated by several conjectures posed in the paper [F. Qi and A.-Q. Liu, Completely monotonic degrees for a difference between the logarithmic and psi functions, J. Comput. Appl. Math. 361 (2019), 366--371; available online at https://doi.org/10.1016/j.cam.2019.05.001], the authors calculate some completely monotonic degrees of remainders of the asymptotic expansion of the logarithm of the gamma function, compute some completely monotonic degrees of remainders of asymptotic expansions of the digamma function, and confirm some conjectures on completely monotonic degrees of remainders of the asymptotic expansion of the logarithm of the gamma function and on completely monotonic degrees of remainders of asymptotic expansions of polygamma functions.

Published
2019

163. A Solution to Qi’s Conjecture on a Double Inequality for a Function Involving the Tri- and Tetra-Gamma Functions

Author

Ladislav Matejíčka

Subjects
Conjecture, trigamma function, biology, General Mathematics, Open problem, Trigamma function, 010102 general mathematics, double optimal inequality, Monotonic function, Function (mathematics), biology.organism_classification, 01 natural sciences, 010101 applied mathematics, Combinatorics, Digamma function, Computer Science (miscellaneous), Tetra, digamma function, 0101 mathematics, Gamma function, Engineering (miscellaneous), Mathematics
Abstract

In the paper, the author gives a solution to a conjecture on a double inequality for a function involving the tri- and tetra-gamma functions, which was first posed in Remark 6 of the paper &ldquo, Complete monotonicity of a function involving the tri- and tetragamma functions&rdquo, (2015) and repeated in the seventh open problem of the paper &ldquo, On complete monotonicity for several classes of functions related to ratios of gamma functions&rdquo, (2019).

Published
2019
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165. On k-analogues of digamma and polygamma functions

Author

Emrah Yildirim and İnci Ege

Subjects
010101 applied mathematics, Pure mathematics, Digamma function, 010102 general mathematics, General Medicine, 0101 mathematics, 01 natural sciences, Mathematics
Published
2018

166. Approximants of the Euler–Mascheroni constant and harmonic numbers.

Author

Burić, Tomislav and Elezović, Neven

Subjects
*APPROXIMATION theory, *EULER-Maclaurin formula, *MATHEMATICAL constants, *HARMONIC analysis (Mathematics), *NUMBER theory, *ASYMPTOTIC expansions, *MATHEMATICAL analysis
Abstract

Abstract: In this paper various approximations for the Euler–Mascheroni constant and harmonic numbers are studied and improved and general asymptotic expansions related to this approximations are obtained. [Copyright &y& Elsevier]

Published
2013
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167. -Extensions of some estimates associated with the digamma function.

Author

Batir, Necdet

Subjects
*ESTIMATION theory, *STATISTICAL association, *GAMMA functions, *MATHEMATICAL inequalities, *MATHEMATICAL analysis, *NUMERICAL analysis
Abstract

Abstract: We extend some known inequalities for the digamma function to the -digamma function. Furthermore, we estimate the unique positive real root of the -digamma function . [Copyright &y& Elsevier]

Published
2013
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168. SOME FUNCTIONAL RELATIONS INVOLVING I-FUNCTION USING FRACTIONAL INTEGRAL OPERATORS.

Author

THAKUR, ANJALI, JAIN, D. K., and JAIN, RENU

Subjects
*HYPERGEOMETRIC functions, *GAMMA functions, *OPERATOR theory, *MATHEMATICAL functions, *INTEGRAL operators
Abstract

There are many papers in mathematical literature in which the functional relation associated with the hypergeometric functions and the logarithmic derivative of gamma function (digamma function) is derived using the application of fractional integration operators. In order to unify and extend the existing results Katsuyuki Nishimoto and R. K. Saxena [16], R. K. Saxena [17], Kalla and Ross [13] and Kalla and Al-saquabi [12] have already contributed their observations. In the proposed paper we seek to establish functional relations between digamma functions and I-function by the application of Erdélyi-Kober integral operators. [ABSTRACT FROM AUTHOR]

Published
2013

169. An approximation formula for n!

Author

Batir, Necdet

Subjects
*APPROXIMATION theory, *MATHEMATICAL formulas, *MATHEMATICAL proofs, *FACTOR analysis, *GAMMA functions, *MATHEMATICAL constants, *MATHEMATICAL inequalities
Abstract

We prove the following very accurate approximation formula for the factorial function: ... This gives better results than the following approximation formula ... which is established by the author [5] and C. Mortici [16] independently, and gives similar results with ... which is established by C. Mortici in his very new paper [8]. [ABSTRACT FROM AUTHOR]

Published
2013
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170. Some infinite sums arising from the Weierstrass product theorem.

Author

Mező, István

Subjects
*INFINITY (Mathematics), *WEIERSTRASS points, *MATHEMATICS theorems, *REPRESENTATION theory, *MATHEMATICAL series, *MATHEMATICAL functions
Abstract

Abstract: A very simple approach using series manipulations and the Weierstrass Representation Theorem yield surprisingly nice and non-trivial series identities involving special functions. In our paper we present a large number of examples of these identities. At the end, several number theoretical sums are deduced. [Copyright &y& Elsevier]

Published
2013
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171. Exponential generating function of hyperharmonic numbers indexed by arithmetic progressions.

Author

Mező, István

Abstract

There is a circle of problems concerning the exponential generating function of harmonic numbers. The main results come from Cvijovic, Dattoli, Gosper and Srivastava. In this paper, we extend some of them. Namely, we give the exponential generating function of hyperharmonic numbers indexed by arithmetic progressions; in the sum several combinatorial numbers (like Stirling and Bell numbers) and the hypergeometric function appear. [ABSTRACT FROM AUTHOR]

Published
2013
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175. Some inequalities for the gamma function

Author

Neuman, Edward

Subjects
*MATHEMATICAL inequalities, *GAMMA functions, *LOGARITHMIC functions, *CONVEX functions, *MATHEMATICS, *MATHEMATICAL analysis
Abstract

Abstract: Several inequalities involving gamma function are obtained. They are established using elementary properties of logarithmically convex functions. [Copyright &y& Elsevier]

Published
2011
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176. The Rényi information in record data from an inverse sampling plan.

Author

Madadi, Mohsen and Tata, Mahbanoo

Subjects
*INVERSE problems, *ZETA functions, *RENYI'S entropy, *INFORMATION theory, *STOCHASTIC convergence
Abstract

In this article we present results on the Rényi information (RI) contained in upper (lower) record values and associated record times in a sequence of i.i.d continuous variables using an inverse sampling plan(ISP). We compute RI for several well known distributions and verify convergence of Rényi information to the relevant Shannon information. A conjecture regarding a characterization of the exponential distribution is also stated. [ABSTRACT FROM AUTHOR]

Published
2011

177. Rational approximations to values of the digamma function and a conjecture on denominators.

Author

Hessami Pilehrood, T. and Hessami Pilehrood, Kh.

Subjects
*APPROXIMATION theory, *GAMMA functions, *LOGARITHMS, *HYPERGEOMETRIC functions, *LAGUERRE polynomials
Abstract

We explicitly construct rational approximations to the numbers ln( b) − ψ( a + 1), where ψ is the logarithmic derivative of the Euler gamma function. We prove formulas expressing the numerators and the denominators of the approximations in terms of hypergeometric sums. This generalizes the Aptekarev construction of rational approximations for the Euler constant γ. As a consequence, we obtain rational approximations for the numbers π/2 ± γ. The proposed construction is compared with rational Rivoal approximations for the numbers γ + ln( b). We verify assumptions put forward by Rivoal on the denominators of rational approximations to the numbers γ + ln( b) and on the general denominators of simultaneous approximations to the numbers γ and ζ(2) − γ. [ABSTRACT FROM AUTHOR]

Published
2011
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178. The quotient of gamma functions by the psi function.

Author

Mortici, Cristinel

Subjects
GAMMA functions, ASYMPTOTIC expansions, MATHEMATICAL series, MATHEMATICAL inequalities, STOCHASTIC convergence, MATHEMATICAL analysis
Abstract

The aim of this paper is to construct the asymptotic series of the ratio of gamma functions by Kershaw then to deduce some sharp estimates. [ABSTRACT FROM AUTHOR]

Published
2011
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179. Refinements of Gurland’s formula for pi

Author

Mortici, Cristinel

Subjects
*GAMMA functions, *MONOTONIC functions, *APPROXIMATION theory, *MATHEMATICAL analysis, *MATHEMATICAL inequalities, *MATHEMATICAL formulas
Abstract

Abstract: The aim of this paper is to refine Gurland’s formula for approximating pi. We prove the complete monotonicity of a class of functions related to Gurland’s formula, and then we state some sharp inequalities. [Copyright &y& Elsevier]

Published
2011
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180. On a Finite Sum Involving Inverse Powers of Cosines.

Author

Kowalenko, Victor

Subjects
*POWER series, *ZETA functions, *APPROXIMATION theory, *NUMBER theory, *NUMERICAL calculations, *MATHEMATICAL formulas, *MATHEMATICAL analysis
Abstract

finite sum S involving inverse powers of cosines has been studied previously by Fisher, who was able to solve the v=1 and v=2 cases exactly and provide the first term of an 'asymptotic solution'. The series is re-visited here by using a completely different approach from Fisher's generating function method. Higher order terms in decreasing powers of m are evaluated in the large m limit. In addition, the exact calculations for the first three integer values of v are presented. An empirical method is then devised, which yields the exact formulae for all the coefficients in S when v is an integer. Consequently, the first ten values of S are tabulated. [ABSTRACT FROM AUTHOR]

Published
2011
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181. Bayesian prediction of order statistics based on k-record values from exponential distribution.

Author

Ahmadi, Jafar, MirMostafaee, S.M.T.K., and Balakrishnan, N.

Subjects
*BAYESIAN analysis, *LOGICAL prediction, *EXPONENTIAL functions, *DISTRIBUTION (Probability theory), *NUMERICAL analysis, *ORDER statistics, *INTERVAL analysis
Abstract

Bayesian prediction of order statistics as well as the mean of a future sample based on observed record values from an exponential distribution are discussed. Several Bayesian prediction intervals and point predictors are derived. Finally, some numerical computations are presented for illustrating all the proposed inferential procedures. [ABSTRACT FROM AUTHOR]

Published
2011
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182. Shannon information in record data.

Author

Madadi, Mohsen and Tata, Mahbanoo

Subjects
*UNCERTAINTY (Information theory), *EULER characteristic, *GAMMA functions, *ZETA functions, *STATISTICAL sampling
Abstract

In this article we present results on the Shannon information (SI) contained in upper (lower) record values and associated record times in a sequence of i.i.d continuous variables. We then establish an interesting relationship between the SI content of a random sample of fixed size, and the SI in the data consisting of sequential maxima. We also consider the information contained in the record data from an inverse sampling plan (ISP). [ABSTRACT FROM AUTHOR]

Published
2011
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183. Complete monotonicity results for some functions involving the gamma and polygamma functions

Author

Şevli̇, Hamdullah and Batır, Necdet

Subjects
*MONOTONE operators, *GAMMA functions, *HARMONIC analysis (Mathematics), *MATHEMATICAL analysis, *MATHEMATICAL programming, *TRANSCENDENTAL functions
Abstract

Abstract: In this paper we prove new complete monotonicity properties of some functions involving the gamma function. As consequences of them we establish various new upper and lower bounds for the gamma function and the harmonic numbers. [Copyright &y& Elsevier]

Published
2011
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184. Transcendence of multi-indexed infinite series

Author

Weatherby, Chester

Subjects
*INFINITE series (Mathematics), *POLYNOMIALS, *PERIODIC functions, *MATHEMATICAL forms, *MATHEMATICAL analysis, *K-theory
Abstract

Abstract: We consider the transcendence of the multi-indexed series and then extend our results to series of the form where f is a k-periodic function and each is a polynomial with algebraic coefficients. We relate these series to special values of the digamma function as well as polygamma functions. We also show some conditional implications on the transcendence of the series via Schanuel''s Conjecture. [Copyright &y& Elsevier]

Published
2011
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185. SHARPNESS OF NEGOI'S INEQUALITY FOR THE EULER-MASCHERONI CONSTANT.

Author

CHAO-PING CHEN

Subjects
*MATHEMATICAL inequalities, *EULER method, *APPROXIMATION theory, *GAMMA functions, *STOCHASTIC convergence
Abstract

We present new estimates for the Euler-Mascheroni constant, which improve a result of Negoi. [ABSTRACT FROM AUTHOR]

Published
2011

186. Sharp inequalities and complete monotonicity for the Wallis ratio.

Author

Mortici, Cristinel

Subjects
*MATHEMATICAL inequalities, *DIFFERENTIAL equations, *MATHEMATICAL functions, *MATHEMATICAL analysis, *MATHEMATICS
Abstract

The aim of this paper is to prove the complete monotonicity of a class of functions arising from Kazarinoff's inequality [Edinburgh Math. Notes 40 (1956) 19-21]. As applications, new sharp inequalities for the gamma and digamma functions are established. [ABSTRACT FROM AUTHOR]

Published
2010
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187. Estimating gamma function by digamma function

Author

Mortici, Cristinel

Subjects
*GAMMA functions, *ESTIMATION theory, *MATHEMATICAL inequalities, *APPROXIMATION theory, *MATHEMATICAL analysis
Abstract

Abstract: The aim of this paper is to refine some recent results stated by Alzer and Grinshpan [H. Alzer, A.Z. Grinshpan, Inequalities for the gamma and -gamma functions J. Approx. Theory 144 (2007) 67–83] and Batir [N. Batir, An interesting double inequality for Euler’s gamma function J. Inequal. Pure Appl. Math. 5(4) (2004) article 97] concerning the problem of approximation of the gamma function in terms of the digamma function. [Copyright &y& Elsevier]

Published
2010
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188. A Refinement of Chen-Qi Inequality on the Harmonic Sum.

Author

Mortici, Cristinel

Subjects
*HARMONIC functions, *HARMONIC analysis (Mathematics), *APPROXIMATION theory, *GAMMA functions, *MATHEMATICAL analysis
Abstract

The aim of this paper is to refine a double inequality of Chen and Qi stated in [The best bounds of harmonic sequence arXiv:math/0306233]. [ABSTRACT FROM AUTHOR]

Published
2010

189. The Dattoli–Srivastava conjectures concerning generating functions involving the harmonic numbers

Author

Cvijović, Djurdje

Subjects
*LOGICAL prediction, *GENERATING functions, *HARMONIC functions, *ANALYTIC functions, *BESSEL functions, *HYPERGEOMETRIC functions
Abstract

Abstract: In a recent paper Dattoli and Srivastava , by resorting to umbral calculus, conjectured several generating functions involving harmonic numbers. In this sequel to their work our aim is to rigorously demonstrate the truth of the Dattoli–Srivastava conjectures by making use of simple analytical arguments. In addition, one of these conjectures is stated and proved in more general form. [Copyright &y& Elsevier]

Published
2010
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190. New approximations of the gamma function in terms of the digamma function

Author

Mortici, Cristinel

Subjects
*APPROXIMATION theory, *GAMMA functions, *ASYMPTOTES, *MATHEMATICAL formulas, *LOGARITHMS, *STOCHASTIC convergence
Abstract

Abstract: The goal of this paper is to prove the following asymptotic formula where is the Euler Gamma function and is the digamma function, namely, the logarithmic derivative of . Moreover, optimal values of parameters are calculated in such a way that this asymptotic convergence is the best possible. [Copyright &y& Elsevier]

Published
2010
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191. Linear independence of digamma function and a variant of a conjecture of Rohrlich

Author

Gun, Sanoli, Ram Murty, M., and Rath, Purusottam

Subjects
*GAMMA functions, *TRANSCENDENTAL functions, *CHI-square distribution, *ALGEBRAIC fields, *ABSTRACT algebra
Abstract

Abstract: Let denote the digamma function. We study the linear independence of at rational arguments over algebraic number fields. We also formulate a variant of a conjecture of Rohrlich concerning linear independence of the log gamma function at rational arguments and report on some progress. We relate these conjectures to non-vanishing of certain L-series. [Copyright &y& Elsevier]

Published
2009
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192. Enumeration by kernel positions

Author

Hetyei, Gábor

Subjects
*KERNEL functions, *COMBINATORIAL enumeration problems, *PARTIALLY ordered sets, *FACTOR analysis, *BERNOULLI numbers, *NEWTON-Raphson method, *GENERATING functions, *BERNOULLI polynomials
Abstract

Abstract: We introduce a class of two-player games on posets with a rank function, in which each move of the winning strategy is unique. This allows one to enumerate the kernel positions by rank. The main example is a simple game on words in which the number of kernel positions of rank n is a signed factorial multiple of the nth Bernoulli number of the second kind. Generalizations to the degenerate Bernoulli numbers and to negative integer substitutions into the Bernoulli polynomials are developed. Using an appropriate scoring system for each function with an appropriate Newton expansion we construct a game in which the expected gain of a player equals the definite integral of the function on the interval . [Copyright &y& Elsevier]

Published
2009
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193. An inequality for ratios of gamma functions

Author

Yu, Yaming

Subjects
*GAMMA functions, *EULER method, *MATHEMATICAL inequalities, *MONOTONIC functions, *SPECIAL functions, *MATHEMATICAL analysis
Abstract

Abstract: Let denote Euler''s gamma function. The following inequality is proved: for and we have The inequality is reversed if . This resolves an open problem of Guo and Qi [B.-N. Guo, F. Qi, Inequalities and monotonicity for the ratio of gamma functions, Taiwanese J. Math. 7 (2003) 239–247]. [Copyright &y& Elsevier]

Published
2009
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195. New inequalities for the Hurwitz zeta function.

Author

Batir, Necdet

Subjects
MATHEMATICS research, ZETA functions, RIEMANNIAN geometry, MATHEMATICAL inequalities, GAMMA functions
Abstract

We establish various new inequalities for the Hurwitz zeta function. Our results generalize some known results for the polygamma functions to the Hurwitz zeta function. [ABSTRACT FROM AUTHOR]

Published
2008
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196. Transcendental values of the digamma function

Author

Ram Murty, M. and Saradha, N.

Subjects
*TRANSCENDENTAL functions, *EULER'S numbers, *SPECIAL functions, *RINGS of integers
Abstract

Abstract: Let denote the digamma function, that is, the logarithmic derivative of Euler''s Γ-function. Let q be a positive integer greater than 1 and γ denote Euler''s constant. We show that all the numbers are transcendental. We also prove that at most one of the numbers is algebraic. [Copyright &y& Elsevier]

Published
2007
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197. On some properties of digamma and polygamma functions

Author

Batir, Necdet

Subjects
*GAMMA functions, *MATHEMATICAL inequalities, *MATHEMATICAL functions, *DIFFERENTIAL equations
Abstract

Abstract: In this note we present some new and structural inequalities for digamma, polygamma and inverse polygamma functions. We also extend, generalize and refine some known inequalities for these important functions. [Copyright &y& Elsevier]

Published
2007
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198. Evaluation of series with Hurwitz and Lerch zeta function coefficients by using Hankel contour integrals

Author

Boyadzhiev, Khristo N.

Subjects
*NUMERICAL functions, *EULER'S numbers, *INTEGRALS, *INTEGRAL calculus
Abstract

Abstract: We introduce a new technique for evaluation of series with zeta coefficients and also for evaluation of certain integrals involving the logGamma function. This technique is based on Hankel integral representations of the Hurwitz zeta, the Lerch Transcendent, the Digamma and logGamma functions. [Copyright &y& Elsevier]

Published
2007
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199. On conditional irrationality measures for values of the digamma function

Author

Hessami Pilehrood, Kh. and Hessami Pilehrood, T.

Subjects
*MATHEMATICAL functions, *WEIGHTS & measures, *ALGEBRA, *MATHEMATICS
Abstract

Abstract: The aim of this work is to obtain the so-called standard lemmas on irrationality bases using the principles of Chudnovsky and then apply them to obtain conditional irrationality measures for values of the digamma function. [Copyright &y& Elsevier]

Published
2007
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200. Several weighted sum formulas of multiple zeta values

Author

Yao Lin Ong, Minking Eie, and Wen-Chin Liaw

Subjects
Discrete mathematics, Algebra and Number Theory, Computer Science::Information Retrieval, 010102 general mathematics, Astrophysics::Instrumentation and Methods for Astrophysics, Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing), 010103 numerical & computational mathematics, 01 natural sciences, Arithmetic zeta function, TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES, Digamma function, ComputingMethodologies_DOCUMENTANDTEXTPROCESSING, Computer Science::General Literature, 0101 mathematics, Special case, Gamma function, Value (mathematics), ComputingMilieux_MISCELLANEOUS, Mathematics, Real number
Abstract

For a real number [Formula: see text] and positive integers [Formula: see text] and [Formula: see text] with [Formula: see text], we evaluate the sum of multiple zeta values [Formula: see text] explicitly in terms of [Formula: see text] and [Formula: see text]. The special case [Formula: see text] gives an evaluation of [Formula: see text]. An explicit evaluation of the multiple zeta-star value [Formula: see text] is also obtained, as well as some applications to evaluation of multiple zeta values with even arguments.

Published
2017

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