LEADER 04601nam 22007093 450 001 996483153603316 005 20220921220714.0 010 $a3-030-95088-3 035 $a(CKB)5700000000101748 035 $a(MiAaPQ)EBC7041855 035 $a(Au-PeEL)EBL7041855 035 $a(OCoLC)1335127471 035 $a(oapen)https://directory.doabooks.org/handle/20.500.12854/87685 035 $a(PPN)263897478 035 $a(EXLCZ)995700000000101748 100 $a20220919d2022 fy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 12$aA generalization of Bohr-Mollerup's theorem for higher order convex functions /$fJean-Luc Marichal, Nai?m Zenai?di 210 $aCham$cSpringer Nature$d2022 210 1$aCham :$cSpringer International Publishing AG,$d2022. 210 4$d©2022. 215 $a1 online resource (xviii, 323 pages) 225 1 $aDevelopments in mathematics$vv.70 311 1 $a3-030-95087-5 327 $aPreface List of main symbols Table of contents Chapter 1. Introduction Chapter 2. Preliminaries Chapter 3. Uniqueness and existence results Chapter 4. Interpretations of the asymptotic conditions Chapter 5. Multiple log-gamma type functions Chapter 6. Asymptotic analysis Chapter 7. Derivatives of multiple log-gamma type functions Chapter 8. Further results Chapter 9. Summary of the main results Chapter 10. Applications to some standard special functions Chapter 11. Defining new log-gamma type functions Chapter 12. Further examples Chapter 13. Conclusion A. Higher order convexity properties B. On Krull-Webster's asymptotic condition C. On a question raised by Webster D. Asymptotic behaviors and bracketing E. Generalized Webster's inequality F. On the differentiability of \sigma_g Bibliography Analogues of properties of the gamma function Index 330 $aIn 1922, Harald Bohr and Johannes Mollerup established a remarkable characterization of the Euler gamma function using its log-convexity property. A decade later, Emil Artin investigated this result and used it to derive the basic properties of the gamma function using elementary methods of the calculus. Bohr-Mollerup's theorem was then adopted by Nicolas Bourbaki as the starting point for his exposition of the gamma function. This open access book develops a far-reaching generalization of Bohr-Mollerup's theorem to higher order convex functions, along lines initiated by Wolfgang Krull, Roger Webster, and some others but going considerably further than past work. In particular, this generalization shows using elementary techniques that a very rich spectrum of functions satisfy analogues of several classical properties of the gamma function, including Bohr-Mollerup's theorem itself, Euler's reflection formula, Gauss' multiplication theorem, Stirling's formula, and Weierstrass' canonical factorization. The scope of the theory developed in this work is illustrated through various examples, ranging from the gamma function itself and its variants and generalizations (q-gamma, polygamma, multiple gamma functions) to important special functions such as the Hurwitz zeta function and the generalized Stieltjes constants. This volume is also an opportunity to honor the 100th anniversary of Bohr-Mollerup's theorem and to spark the interest of a large number of researchers in this beautiful theory. 410 0$aDevelopments in mathematics$v70. 606 $aConvex functions 606 $aGamma functions 610 $aDifference Equation 610 $aHigher Order Convexity 610 $aBohr-Mollerup's Theorem 610 $aPrincipal Indefinite Sums 610 $aGauss' Limit 610 $aEuler Product Form 610 $aRaabe's Formula 610 $aBinet's Function 610 $aStirling's Formula 610 $aEuler's Infinite Product 610 $aEuler's Reflection Formula 610 $aWeierstrass' Infinite Product 610 $aGauss Multiplication Formula 610 $aEuler's Constant 610 $aGamma Function 610 $aPolygamma Functions 610 $aHurwitz Zeta Function 610 $aGeneralized Stieltjes Constants 615 0$aConvex functions. 615 0$aGamma functions. 700 $aMarichal$b Jean-Luc$01255007 701 $aZenaïdi$b Naïm$01255008 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a996483153603316 996 $aA generalization of Bohr-Mollerup's theorem for higher order convex functions$92909870 997 $aUNISA