A BIBLIOGRAPHY OF GAMMA FUNCTION AND ...

A BIBLIOGRAPHY OF GAMMA FUNCTION AND RELATED TOPICS Version 0.5. – October 2011, http://milanmerkle.com The original entries in this bibliography were created by merging the bibliography of J´ ozsef S´ andor with mine, resulting in the Version 0.4 with total of 892 items. Every item entered after Version 0.4 is accompanied by a note that contains the name of a person who submitted it and the date of submission. Milan Merkle

References [1] Ibrahim A. Abou Tair. On certain Dirichlet series related to Hurwitz Zeta-function. J. Inst. Math. Comput. Sci. Math. Ser. [Journal of Institute of Mathematics and Computer Sciences (Mathematics Series)] 3(3):299–304, 1990. [2] Jorge Alberto Achcar, Heleno Bolfarine. The log-linear model with a generalized Gamma distribution for the error: A Bayesian approach. Statist. Probab. Lett. [Statistics and Probability Letters] 4(6):325–332, 1986. [3] V. S. Adamchik, O. I. Marichev. Representations of functions of hypergeometric type in logarithmic cases. Vestsi Akad. Navuk BSSR Ser. Fiz. Mat. Navuk [Vestsi Akademii Navuk BSSR. Seryya Fizika Matematychnykh Navuk](5):29–35, 1983. In Russian. [4] A. Adatia, A. G. Law, Q. Wang. Characterization of a mixture of Gamma distributions via conditional finite moments. Comm. Statist. Theory Methods [Communications in Statistics. Theory and Methods] 20(5–6):1937–1949, 1991. [5] I. I. Adgamov, I. N. Volodin. On a test for the Weibull distribution against a family of generalized Gamma-alternatives. Izv. Vyssh. Uchebn. Zaved. Mat. [Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika](6):3–8, 90, 1987. Russian. [6] C. Adiga, T. Kim. On a generalization of S´ andor’s function. Proc. Jangjeon Math. Soc. 5(2):121–124, 2002. [7] C. Adiga et al. On a q-analogue of S´ andor’s function. J. Ineq. Pure Appl. Math. 4(5), 2003. art.84 (electronic). [8] Alan Adolphson. Uniqueness of γp in the Gross-Koblitz formula for Gauss sums. Trans. Amer. Math. Soc. [Transactions of the American Mathematical Society] 278(1):57–63, 1983. [9] A.U. Afuwape, C.O. Imoru. Bounds for the Beta function. Bolletino U.M.I. 5(17-A):330– 334, 1980. [10] R.P. Agarwal. Difference equations and inequalities. Marcel Dekker, Inc., 2nd edition, 2000. [11] Abdul Hadi Nabih Ahmed, A. M. Abouammoh. Characterizations of Gamma, inverse Gaussian, and negative binomial distributions via their length-biased distributions. Statist. Papers [Statistical Papers. Statistische Hefte] 34(2):167–173, 1993. [12] A. M. Al Rashed, S. I. Ahmed. A generalization of the number π. J. Natur. Sci. Math. [The Journal of Natural Sciences and Mathematics] 29(1):29–37, 1989. [13] M. Masoom Ali, A. K. Md. E. Saleh, Dale Umbach. Estimating functions of location and scale parameters. Soochow J. Math. [Soochow Journal of Mathematics] 19(3):259–270, 1993. [14] A. Alikhani, M. Hassani. Approximation of pn by hn . RGMIA Research Report Collection 8(4), 2005. [15] Giampietro Allasia, Renata Besenghi. Numerical calculation of incomplete gamma functions by the trapezoidal rule. Numer. Math. [Numerische Mathematik] 50(4):419–428, 1987. [16] Giampietro Allasia, Renata Besenghi. Numerical calculation of the Gamma and Digamma functions using the trapezoidal rule. Boll. Un. Mat. Ital. B (7) [Unione Matematica Italiana. Bollettino. B. Serie-VII] 1(3):815–828, 1987. Italian. [17] J.P. Allouche. Transcendence of the Carlitz-Goss Gamma function at rational arguments. J. Number Theory 60(2):318–328, 1996. [18] C. Alsina, M.S. Tom´ as. A geometrical proof of a new inequality for the Gamma function. J. Ineq. Pure Appl. Math. 6(2), 2005. 1

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Bibliography of Gamma function–v.0.5 (October 2011)

http://milanmerkle.com

[19] A.A. Alyakrinskii. On the representation of the values of Euler’s Gamma function at some rational points in the form of an infinite product. In Investigations in complex analysis (Russian), 123–128. Krasnoyarsk. Gos. Univ., 1989. [20] H. Alzer. On some inequalities involving (n!)1/n , II. Period. Math. Hung. 28(3):229–233, 1994. [21] H. Alzer. Note on an inequality involving (n!)1/n . Acta Math. Univ. Comen. New Ser. 64:283–285, 1995. [22] H. Alzer. Characterizations of the ratio of Gamma and q-Gamma functions. Abh. Math. Sem. Univ. Hamburg 70:165–174, 2000. [23] H. Alzer. Inequalities for the Gamma function. Proc. Amer. Math. Soc. 128:141–147, 2000. [24] H. Alzer. A mean-value inequality for the Gamma function. Applied Math. Letters 13:111– 114, 2000. [25] H. Alzer. A power mean inequality for the Gamma function. Monatsh. Math. 131:179–188, 2000. [26] H. Alzer. Sharp bounds for the ratio of q-Gamma functions. Math. Nachr. 222:5–14, 2001. [27] H. Alzer. Sharp inequalities for the Beta function. Indag Math. (N.S.) 12:15–21, 2001. [28] H. Alzer. Inequalities for the Beta function of n variables. The ANZIAM Journal 44:609– 623, 2003. [29] H. Alzer. On Ramanujan’s double inequality for the Gamma function. Bull. London Math. Soc. 35:601–607, 2003. [30] H. Alzer. A superadditive property of hadamard’s gamma function. Abh.Math.Semin.Univ. Hambg. 79:11–23, 2009. [31] H. Alzer, D. Karayannakis, H. M. Srivastava. Series representations for some mathematical constants. J. Math. Anal. Appl. 320:145–162, 2006. Submitted by. H. M. Srivastava, 2. May 2010. [32] H. Alzer, S. Ruscheweyh. A subadditive property of the Gamma function. J. Math. Anal. Appl. 285:564–577, 2003. [33] Horst Alzer. Some Gamma function inequalities. Math. Comp. [Mathematics of Computation] 60(201):337–346, 1993. [34] Horst Alzer, Jim Wells. Inequalities for the Polygamma functions. SIAM J. Math. Anal 29(6):1459–1466, 1998. [35] Y. Amice, G. Christol, P. Robba, editors. Groupe d’etude d’analyse ultrametrique. 9e annee: 1981/82. Fasc. 1. [Study Group on Ultrametric Analysis. 9th year: 1981/82. No. 1]. Institut Henri Poincare, 115 pp., Paris, 1983. French. [36] Y. Amice, G. Christol, P. Robba, editors. Groupe d’etude d’analyse ultrametrique. 9e annee: 1981/82. Fasc. 3. [Study Group on Ultrametric Analysis. 9th year: 1981/82. No. 3]. Institut Henri Poincare, 125 pp, Paris, 1983. Conference: p-adic analysis and its applications; September 6–10, Marseille, 1982; Edited by Y. Amice, In French. [37] Yvette Amice. Fonction γ p-adique associ´ ee a un caractere de Dirichlet. [p-adic γ function associated with a Dirichlet character]. In Collection: Study Group on Ultrametric Analysis. 7th–8th years: 1979–1981 (Paris, 1979/1981) (French), Exp. No. 17, 11 pp, Paris, 1981. Secretariat Math. French. [38] M. Amou. Irrationality results for values of certain q-functions. Proc. Jangjeon Math. Soc.:27–36, 2000. [39] J. Anastassiadis. D´ efinition des fonctions eul´ eriennes par des ´ equations fonctionnelles. Gauthier-Villars, Paris, 1964. [40] Jean Anastassiadis. Fonctions semi-monotones et semi-convexes et solutions d’une ´ equation fonctionnelle. Bull. Sci. Math, 2e serie 76:148–160, 1952. French. [41] Jean Anastassiadis. Sur les solutions logarithmiquement convexes ou concaves d’une ´ equation fonctionnelle. Bull. Sci. Math, 2e serie 81:78–87, 1952. French. [42] Jean Anastassiadis. Une propri´ et´ e de la fonction Gamma. Bull. Sci. Math, 2e serie 81:116– 118, 1957. French. [43] Jean Anastassiadis. Definitions fonctionnelles de la Fonction b(x, y). Bull. Sci. Math, 2e serie 83:24–32, 1959. French. [44] Jean Anastassiadis. Remarques sur les quelques ´ equations fonctionnelles. C. R. Acad. Sci. Paris 250:2663–2665, 1960. French. [45] Jean Anastassiadis. Sur les solutions de l’´ equation fonctionnelle f (x + 1) = ϕ(x)f (x). C. R. Acad. Sci. Paris 253:2446–2447, 1961. French.


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[46] G.D. Anderson, M.K. Vamanamurthy, M. Vuorinen. Conformal invariants, inequalities and quasiconformal maps. Wiley, New York, 1997. [47] G.D. Anderson, M.K. Vamanamurthy, M. Vuorinen. Topics in special functions. In Papers on Analysis, a volume dedicated to Olli Martio, Report Univ. Jyv¨ askyl¨ a, volume 83, 5–26, 2001. [48] Greg W. Anderson. Logarithmic derivatives of Dirichlet l-functions and the periods of Abelian varieties. Compositio Math. [Compositio Mathematica] 45(3):315–332, 1982. [49] Greg W. Anderson. Cyclotomy and an extension of the Taniyama group. Compositio Math. [Compositio Mathematica] 57(2):153–217, 1986. [50] Greg W. Anderson. The hyperadelic Gamma function: A precis. In Collection: Galois representations and arithmetic algebraic geometry (Kyoto, 1985/Tokyo, 1986), volume 12 of Adv. Stud. Pure Math, 1–19. North-Holland, Amsterdam-New York, 1987. [51] Greg W. Anderson. The hyperadelic Gamma function. Invent. Math. [Inventiones Mathematicae] 95(1):63–131, 1989. [52] Greg W. Anderson. Normalization of the hyperadelic Gamma function. In Collection: Galois groups over Q (Berkeley, CA, 1987), volume 16 of Math. Sci. Res. Inst. Publ, 1–31. Springer, New York-Berlin, 1989. [53] G.E. Andrews. q-Series: Their development and applications in Analysis, Number theory, Combinatorics, Physics and Computer algebra. Chelsea, USA, 1986. reprinted 1991. [54] G.E. Andrews, R. Askey. Another q-extension of the Beta function. Proc. A.M.S. 81:97–100, 1981. [55] G.E. Andrews, R. Askey. Another q-extension of the Beta function. Proc. AMS 81(1):97– 100, 1981. [56] G.E. Andrews, R. Askey, R. Roy. Special functions. Cambridge Univ. Press, 1999. [57] P. Appell. Sur un classe de fonctions analogues aux fonctions Eul´ eriennes ´ etudi´ ees par M. Heine. C.R. Acad. Sci. Paris 89:841–844, 1979. [58] T. Artikis. Convex Densities and Self-Decomposability. Serdica [Serdica. Bulgaricae Mathematicae Publicationes] 9(3):326–329, 1983. [59] E. Artin. Einf¨ uhrung in die theorie der Gammafunktion. B.G. Teubner, Berlin-Leipzig, 1931. [60] Emil Artin. The Gamma Function. Holt, Rinehart and Winston, New York, 1964. Translation from the German Original of 1931. [61] R. Askey. A q-extension of Cauchy’s form of the Beta integral. Quart. J. Math. Oxford Ser. (2) 32(127):255–266, 1981. [62] Richard Askey. The q-Gamma and qz-Beta functions. Applicable Anal. [Applicable Analysis. An International Journal] 8(2):125–141, 1978–1979. [63] Richard Askey. Ramanujan’s extensions of the Gamma and Beta functions. Amer. Math. Monthly [The American Mathematical Monthly] 87(5):346–359, 1980. [64] J. P. Asselin de Beauville. Estimation non parametrique de la densite et du mode exemple de la distribution Gamma. Rev. Statist. Appl. [Revue de Statistique Appliquee. Centre d’Enseignement et de Recherche de Statistique Appliquee] 26(3):47–70, 1978. French. [65] A. C. Atkinson. The simulation of generalized inverse Gaussian and hyperbolic random variables. SIAM J. Sci. Statist. Comput. [Society for Industrial and Applied Mathematics. Journal on Scientific and Statistical Computing] 3(4):502–515, 1982. [66] A. M. Awad, M. M. Azzam, M. A. Hamdan. Power series mixture of Gamma distribution. Iraqi J. Sci. [Iraqi Journal of Science] 22(1):108–121, 1981. [67] Adnan M. Awad. On noncentral distribution and approximations to distributions of the randomized Gamma type. Dirasat Res. J. Natur. Sci. [Dirasat. A Research Journal. Natural Sciences] 2(1):17–34, 1975. [68] K.-I. Babenko. Addendum: “The maximum principle for the Euler-Tricomi equation”. Dokl. Akad. Nauk SSSR [Doklady Akademii Nauk SSSR] 287(3):520, 1986. [69] M. Badiale. Some characterization of the q-gamma function by functional equations I. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. (8) 74(1):7–11, 1983. [70] M. Badiale. Some characterization of the q-gamma function by functional equations II. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. (8) 74(2):49–54, 1983. [71] Marino Badiale. Some characterisation of the q-Gamma function by functional equations. I. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur 8, 1983. [72] Marino Badiale. Some characterization of the q-Gamma function by functional equations. II. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur 8, 1983.

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[73] Marino Badiale, Francis J. Sullivan. A note on Badiale’s characterization of the q-Gamma functions. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. [Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisic] 8, 1984. [74] D.H. Bailey. Numerical results on the transcendence of constants involving π, e, and Euler’s constant. Math. Comp. 50(181):275–281, 1988. [75] W.N. Bailey. Generalized hypergeometric series. Cambridge Univ. Press, 1935. In series: Cambridge Tracts in Mathematics (Cambridge Math. Tract.), No. 32. [76] Francesco Baldassarri. Higher p-adic Gamma functions and Dwork cohomology. Asterisque 4(119–120):111–127, 1984. P -adic cohomology. [77] S.B. Bank. Some results on the Gamma function and other hypertranscendental functions. Proc. Royal Soc. Edinb. Sect. A 79(3–4):335–341, 1977/78. [78] Steven B. Bank. A note on zeros of differential polynomials in the Gamma function. Comment. Math. Univ. St. Paul. [Commentarii Mathematici Universitatis Sancti Pauli] 32(1):77–84, 1983. [79] Steven B. Bank, Robert P. Kaufman. On the Gamma function and the Nevanlinna characteristic. Analysis [Analysis. International Journal of Analysis and its Application] 6(2– 3):115–133, 1986. [80] G. B´ ank¨ ovi. A decomposition-rejection technique for generating exponential random variables. Math. Proc. of the Hung. Acad. of Science, Ser. A 9:573–581, 1964. [81] G. B´ ank¨ ovi. A note on the generation of Beta distributed and Gamma distributed random variables. Math. Proc. of the Hung. Acad. of Science, Ser. A 9:555–562, 1964. [82] Shaul K. Bar Lev, Benjamin Reiser. A note on maximum conditional likelihood estimation for the Gamma distribution. Sankhya Ser. B [Sankhya (Statistics). The Indian Journal of Statistics. Series B] 45(2):300–302, 1983. [83] Mahmud Barjaktarevi´ c. O rjeˇsenjima jedne funkcionalne jednaˇ cine (On Solutions of a Functional Equation). Glasnik Mat. Fiz. Astronom 8:297–300, 1953. Serbian. [84] E.W. Barnes. The theory of the G-function. Quart. J. Math. 31:264–314, 1899. [85] E.W. Barnes. The theory of the double Gamma function. Philos. Trans. Roy. Soc. London Ser. A 196:265–388, 1901. [86] E.W. Barnes. On the theory of the multiple Gamma-function. Cambr. Trans. 19:374–425, 1904. [87] Daniel Barsky. On Morita’s p-adic γ-function. In Collection: Groupe d’Etude d’Analyse Ultrametrique, 5e annee (1977/78), Exp. No. 3, 6 pp., Paris, 1978. Secretariat Math. [88] Daniel Barsky. On Morita’s p-adic Gamma function. Math. Proc. Cambridge Philos. Soc. [Mathematical Proceedings of the Cambridge Philosophical Society] 89(1):23–27, 1981. [89] J. Bastero, F. Galve, A. Pe˜ na, M. Romano. Inequalities for the Gamma function and estimates for the volume of sections of bn p . Proc. A.M.S. 130(1):183–192, 2001. [90] G. Bastien, M. Rogalski. Convexit´ e, complete monotonie et inegalit´ es sur les fonctions Zeta et Gamma sur les fonctions des operateurs de Baskarov et sur des fonctions arithmetiques. Canad. J. Math. 54(5):916–944, 2002. [91] Jesus Basulto. Square root versus logarithmic transformation of a Gamma distribution. Trabajos Estadist. Investigacion Oper. [Trabajos de Estadistica y de Investigacion Operativa] 29(3):83–88, 1978. Spanish. [92] H. Bateman, A. Erd´ elyi. Higher transcendental functions. McGraw-Hill, New York, 1955. [93] N. Batir. An interesting double inequality for Euler’s Gamma function. J. Ineq. Pure Appl. Math. 5(4), 2004. [94] N. Batir. Some new inequalities for Gamma and Polygamma functions. J. Ineq. Pure Appl. Math. 6(4), 2005. [95] N. Batir. Sharp inequalities for factorial n. RGMIA Research Report Collection 9(3), 2006. [96] N. Batir. Some Gamma function inequalities. RGMIA Research Report Collection 9(3), 2006. [97] E.F. Beckenbach, R. Bellman. Inequalities. Springer Verlag, 4th edition, 1983. first ed., 1961. [98] P. J. Becker, J. J. J. Roux. A bivariate extension of the Gamma distribution. South African Statist. J. [South African Statistical Journal] 15(1):1–12, 1981. [99] John Beebee. Exact covering systems and the Gauss-Legendre multiplication formula for the Gamma function. Proc. Amer. Math. Soc. [Proceedings of the American Mathematical Society] 120(4):1061–1065, 1994.



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[100] P.R. Beesack. Inequalities for absolute moments of a distribution: From Laplace to von Mises. J. Math. Anal. Appl. 98:435–457, 1984. [101] M. I. Beg. On estimation of scale parameter in Gamma distribution. IAPQR Trans. [IAPQR Transactions. Journal of the Indian Association for Productivity, Quality and Reliability] 12(1):83–85, 1987. [102] Bradley M. Bell. Generalized Gamma parameter estimation and moment evaluation. Comm. Statist. Theory Methods [Communications in Statistics. Theory and Methods] 17(2):507– 517, 1988. [103] M. Bencze. New generalization of the convexity (concavity). Octogon Math. M. 10(1):43–50, 2002. [104] M. Bencze. Special inequalities for increasing and convex functions. Octogon Math. M. 10(1):296–321, 2002. [105] M. Bencze. An inequality for convex functions. Octogon Math. M. 11(2):504–507, 2003. [106] M. Bencze. Inequalities for Gamma function. Octogon Math. M. 13(2):982–984, 2005. [107] M. Bencze, D.M. B˘ atinet¸u-Giurgiu. Inequalities for Gamma function. Octogon Math. Mag. 10(1):233–246, 2002. [108] M. Bencze, C. J. Zhao. The lambda-multiplicatively convex (concave) version of Hadamard’s inequality. Octogon Math. M. 10(1):93–102, 2002. [109] L. Bendersky. Sur la fonction Gamma g´ en´ eralis´ ee. Acta Math. 61:263–322, 1933. [110] C. Berg. Integral representation of some functions related to the Gamma function. Mediterr. J. Math. 1:433–439, 2004. Submitted by. Christian Berg, 4. May 2010. [111] C. Berg. On a generalized Gamma convolution related to the q-calculus. In M. E. H. Ismail, Erik Koelink, editors, Developments of Mathematics, ”Theory and Applications of Special Functions”. A volume dedicated to Mizan Rahman., volume 13, 61–76. Springer Science+ Business Media, Inc., Berlin, 2005. Submitted by. Christian Berg, 4. May 2010, ISBN: 0387-24231-7. [112] C. Berg, G. Forst. Potential theory on locally compact Abelian groups. Springer-Verlag, Berlin, 1975. [113] C. Berg, H. L. Pedersen. A Pick function related to the sequence of volumes of the unit ball in n-space. Submitted by. Christian Berg, 4. May 2010, (ArXiv 0912.2185) (Manuscript from December). [114] C. Berg, H. L. Pedersen. A completely monotone function related to the Gamma function. J. Comput. Appl. Math. 133:219–230, 2001. Submitted by. Christian Berg, 4. May 2010. [115] C. Berg, H. L. Pedersen. Pick functions related to the Gamma function. Rocky Mountain Journal of Mathematics 32(2):507–525, 2002. Submitted by. Christian Berg, 4. May 2010. [116] C. Berg, H. L. Pedersen. The Chen-Rubin conjecture in a continuous setting. Methods and Applications of Analysis 13(1):63–88, 2006. Submitted by. Christian Berg, 4. May 2010. [117] C. Berg, H. L. Pedersen. Convexity of the median in the Gamma distribution. Ark. Mat. 46:1–6, 2008. Submitted by. Christian Berg, 4. May 2010. [118] A. Berger. Sur quelques applications de la fonction Gamma a ` la th´ eorie des nombres. Akad. Afhandt., Uppsala, 1880. [119] Mark Berman. The maximum likelihood estimators of the parameters of the Gamma distribution are always positively biased. Comm. Statist. A. Theory Methods [Communications in Statistics. A. Theory and Methods] 10(7):693–697, 1981. [120] B.C. Berndt. The Gamma function and the Hurwitz Zeta function. Amer. Math. Monthly 92:126–130, 1985. [121] B.C. Berndt, J. Sohn. Asymptotic formulas for two continued fractions in Ramanujan’s lost notebook. J. London Math. Soc. 2(65):271–284, 2002. [122] Bruce C. Berndt. Chapter 8 of Ramanujan’s 2nd notebook. J. Reine Angew. Math. [Journal fur die Reine und Angewandte Mathematik] 338:1–55, 1983. [123] Bruce C. Berndt. The Gamma function and the Hurwitz Zeta-function. Amer. Math. Monthly [The American Mathematical Monthly] 92(2):126–130, 1985. [124] Bruce C. Berndt. Ramanujan’s notebooks. Part I. Springer-Verlag, x+357 pp, $ 54.00., New York-Berlin, 1985. With a foreword by S. Chandrasekhar. [125] Bruce C. Berndt, Robert L. Lamphere, B. M. Wilson. Chapter 12 of Ramanujan’s 2nd notebook: Continued fractions. Rocky Mountain J. Math. [The Rocky Mountain Journal of Mathematics] 15(2):235–310, 1985. Number theory (Winnipeg, Man, 1983).

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