Advanced search
Publication Cover
Applicable Analysis
An International Journal
Volume 8, 1978 - Issue 2
Submit an article Journal homepage
319
Views
155
CrossRef citations to date
0
Altmetric
Original Articles

The q-Gamma and q-Beta FunctionsFootnote†

Richard Askey University of Wisconsin, Department of Mathematics, Van Vleck Hall, Madison, Wisconsin, 53706, USA
Pages 125-141 | Received 16 Sep 1977, Published online: 10 May 2007

Citations (155)

Keep up to date with the latest research on this topic with citation updates for this article.

Subscribe to citation updates

Read on this site (5)

Wolfram Koepf, Predrag M. Rajković & Sladjana D. Marinković. (2016) On a connection between formulas about q–gamma functions. Journal of Nonlinear Mathematical Physics 23:3, pages 343-350.
Read now
Jian-Ping Fang. (2015) Remarks on a generalized q-difference equation. Journal of Difference Equations and Applications 21:10, pages 934-953.
Read now
M Mansour. (2006) An asymptotic expansion of the q-gamma function Γq(x). Journal of Nonlinear Mathematical Physics 13:4, pages 479-483.
Read now
S. Bhargava & C. Adiga. (1994) A basic bilateral series summation formula and its applications. Integral Transforms and Special Functions 2:3, pages 165-184.
Read now
Richard Askey. (1980) Ramanujan's Extensions of the Gamma and Beta Functions. The American Mathematical Monthly 87:5, pages 346-359.
Read now

Articles from other publishers (150)

Miguel Vivas–Cortez, Muhammad Zakria Javed, Muhammad Uzair Awan, Muhammad Aslam Noor & Silvestru Sever Dragomir. (2024) Bullen-Mercer type inequalities with applications in numerical analysis. Alexandria Engineering Journal 96, pages 15-33.
Crossref
Mohammad Younus Bhat, Humaira Maqbool & Altaf A. Bhat. (2024) q-Difference Recurrence Relations of Aleph Function with Generalization to nth Derivative. International Journal of Applied and Computational Mathematics 10:2.
Crossref
Nada K. Mahdi & Ayad R. Khudair. (2023) An analytical method for fractional dynamical equations on time scales . Partial Differential Equations in Applied Mathematics 8, pages 100585.
Crossref
Fatma Hıra. (2023) Hahn Laplace transform and its applications. Demonstratio Mathematica 56:1.
Crossref
Mohammed Fadel, Nusrat Raza & Wei-Shih Du. (2023) Characterizing q-Bessel Functions of the First Kind with Their New Summation and Integral Representations. Mathematics 11:18, pages 3831.
Crossref
Nada K. Mahdi & Ayad R. Khudair. (2023) Stability of nonlinear -fractional dynamical systems on time scale . Partial Differential Equations in Applied Mathematics 7, pages 100496.
Crossref
D.J. Booth, I.M. Griffiths & P.D. Howell. (2023) Circular bubbles in a Hele-Shaw channel: a Hele-Shaw Newton's cradle. Journal of Fluid Mechanics 954.
Crossref
V. Rexma Sherine, T. G. Gerly, P. Chellamani, Esmail Hassan Abdullatif Al-Sabri, Rashad Ismail, G. Britto Antony Xavier & N. Avinash. (2022) A Method for Performing the Symmetric Anti-Difference Equations in Quantum Fractional Calculus. Symmetry 14:12, pages 2604.
Crossref
Gurhan Icoz & Seda Demir. (2022) Approximation Properties of a New Type of Gamma Operator Defined with the Help of -Gamma Function . Journal of Function Spaces 2022, pages 1-9.
Crossref
Horst Alzer & Ahmed Salem. (2021) A harmonic mean inequality for the q-gamma function. The Ramanujan Journal 58:4, pages 1025-1041.
Crossref
H. ALZER & A. SALEM. (2022) FUNCTIONAL INEQUALITIES FOR THE q-DIGAMMA FUNCTION. Acta Mathematica Hungarica 167:2, pages 561-575.
Crossref
Khaled Mehrez & Sourav Das. (2022) Logarithmically completely monotonic functions related to the q-gamma function and its applications. Analysis and Mathematical Physics 12:2.
Crossref
Yasir Nawaz, Muhammad Shoaib Arif, Kamaleldin Abodayeh & Mairaj Bibi. (2022) Finite difference schemes for time-dependent convection <i>q</i>-diffusion problem. AIMS Mathematics 7:9, pages 16407-16421.
Crossref
Jean-Luc Marichal & Naïm ZenaïdiJean-Luc Marichal & Naïm Zenaïdi. 2022. A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions. A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions 179 254 .
Ying Sheng & Tie Zhang. (2021) Some Results on the q-Calculus and Fractional q-Differential Equations. Mathematics 10:1, pages 64.
Crossref
Faris Alzahrani, Ahmed Salem & Moustafa El-Shahed. (2021) Sharp bounds for a ratio of the q-gamma function in terms of the q-digamma function. Journal of Inequalities and Applications 2021:1.
Crossref
Christian Berg, Asena Çetinkaya & Dmitrii Karp. (2021) Completely monotonic ratios of basic and ordinary gamma functions. Aequationes mathematicae 95:3, pages 569-588.
Crossref
TIE ZHANG & YUZHONG WANG. (2021) THE UNIQUE EXISTENCE OF SOLUTION IN THE q-INTEGRABLE SPACE FOR THE NONLINEAR q-FRACTIONAL DIFFERENTIAL EQUATIONS. Fractals 29:03, pages 2150050.
Crossref
Nusrat Raza, Mohammed Fadel, Kottakkaran Sooppy Nisar & M. Zakarya. (2021) On 2-variable $ q $-Hermite polynomials. AIMS Mathematics 6:8, pages 8705-8727.
Crossref
Sahar Zafari, Mariia Murashkina, Tuomas Eerola, Jouni Sampo, Heikki Kälviäinen & Heikki Haario. (2020) Resolving overlapping convex objects in silhouette images by concavity analysis and Gaussian process. Journal of Visual Communication and Image Representation 73, pages 102962.
Crossref
Mohamed El Bachraoui & József Sándor. (2020) Some new sums of q-trigonometric and related functions through a theta product of Jacobi. International Journal of Number Theory 16:08, pages 1803-1817.
Crossref
TIE ZHANG & CAN TONG. (2020) A DIFFERENCE METHOD FOR SOLVING THE NONLINEAR q-FRACTIONAL DIFFERENTIAL EQUATIONS ON TIME SCALES. Fractals 28:06, pages 2050121.
Crossref
Tie Zhang & Qingxin Guo. (2020) The solution theory of the nonlinear -fractional differential equations . Applied Mathematics Letters 104, pages 106282.
Crossref
Pin Lyu & Seakweng Vong. (2020) An efficient numerical method for -fractional differential equations . Applied Mathematics Letters 103, pages 106156.
Crossref
Mohamed El Bachraoui & József Sándor. (2020) On q-Raabe integral and some extensions. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 114:2.
Crossref
Tie Zhang & Yongchao Tang. (2019) A difference method for solving the -fractional differential equations . Applied Mathematics Letters 98, pages 292-299.
Crossref
Juan Pablo Vigneaux. (2019) Information Theory With Finite Vector Spaces. IEEE Transactions on Information Theory 65:9, pages 5674-5687.
Crossref
Rouven Frassek. (2019) The non-compact XXZ spin chain as stochastic particle process. Journal of Physics A: Mathematical and Theoretical 52:33, pages 335202.
Crossref
Yongchao Tang & Tie Zhang. (2019) A remark on the q-fractional order differential equations. Applied Mathematics and Computation 350, pages 198-208.
Crossref
Ahmed Salem. (2018) Sharp bounds for the q-gamma function in terms of the Lambert W function. The Ramanujan Journal 49:2, pages 321-339.
Crossref
Sourav Das. (2017) Inequalities for q-gamma function ratios. Analysis and Mathematical Physics 9:1, pages 313-321.
Crossref
Mohamed El Bachraoui & James Griffin. (2017) Reciprocity theorems involving the q-gamma function. The Ramanujan Journal 45:3, pages 683-694.
Crossref
Sid-Ahmed Yahiaoui & Mustapha Bentaiba. (2018) New Fourier transform containing a pair of complex Euler Γ-functions with a monomial: Mathematical and physical applications. Journal of Mathematical Physics 59:2.
Crossref
Ahmed Salem & Faris Alzahrani. (2017) Two-Sided Inequality Involving the q-Gamma Function. International Journal of Applied and Computational Mathematics 4:1.
Crossref
Shovan Dutta & Erich J. Mueller. (2017) Collective Modes of a Soliton Train in a Fermi Superfluid. Physical Review Letters 118:26.
Crossref
Mohamed El Bachraoui. (2017) Short proofs for q-Raabe formula and integrals for Jacobi theta functions. Journal of Number Theory 173, pages 614-620.
Crossref
Zeinab Sayed Mansour & Maryam Ahmed Al-Towailb. (2016) A reduction formula for a q-beta integral. Advances in Difference Equations 2016:1.
Crossref
M. K. Kerimov. (2016) Studies on the zeros of Bessel functions and methods for their computation: 3. Some new works on monotonicity, convexity, and other properties. Computational Mathematics and Mathematical Physics 56:12, pages 1949-1991.
Crossref
Kwara Nantomah & Edward Prempeh. (2016) INEQUALITIES FOR THE (q, k)-DEFORMED GAMMA FUNCTION EMANATING FROM CERTAIN PROBLEMS OF TRAFFIC FLOW. Honam Mathematical Journal 38:1, pages 9-15.
Crossref
Ahmed Salem. (2015) Monotonic functions related to the q-gamma function. Monatshefte für Mathematik 179:2, pages 281-292.
Crossref
Hédi Elmonser. (2015) Some proprieties for the q-Gamma functions via Wielandt's Theorem. Journal of Number Theory 156, pages 331-339.
Crossref
Ahmed SALEM & Eid S. KAMEL. (2015) Some completely monotonic functions associated with the q -gamma and the q -polygamma functions. Acta Mathematica Scientia 35:5, pages 1214-1224.
Crossref
YOUNG-KI CHO, TAEKYUN KIM, TOUFIK MANSOUR & SEOG-HOON RIM. (2015) On a (r, s)-Analogue of Changhee and Daehee Numbers and Polynomials. Kyungpook mathematical journal 55:2, pages 225-232.
Crossref
Necdet Batir. (2015) Monotonicity properties of -digamma and -trigamma functions . Journal of Approximation Theory 192, pages 336-346.
Crossref
Jian-Ping Fang. (2014) Applications of a generalized q-difference equation. Advances in Difference Equations 2014:1.
Crossref
Won Sang Chung, Taekyun Kim & Toufik Mansour. (2014) THE q-DEFORMED GAMMA FUNCTION AND q-DEFORMED POLYGAMMA FUNCTION. Bulletin of the Korean Mathematical Society 51:4, pages 1155-1161.
Crossref
Ahmed Salem. (2013) An infinite class of completely monotonic functions involving the -gamma function . Journal of Mathematical Analysis and Applications 406:2, pages 392-399.
Crossref
Necdet Batir. (2013) -Extensions of some estimates associated with the digamma function . Journal of Approximation Theory 174, pages 54-64.
Crossref
A N F Aleixo & A B Balantekin. (2013) Generalized coherent, squeezed and intelligent states for exactly solvable quantum systems and the analogue of the displacement and squeezing operators. Journal of Physics A: Mathematical and Theoretical 46:31, pages 315303.
Crossref
V.B. Krasniqi, H.M. Srivastava & S.S. Dragomir. (2013) Some complete monotonicity properties for the -gamma function. Applied Mathematics and Computation 219:21, pages 10538-10547.
Crossref
Ahmad El-Guindy & Zeinab Mansour. (2012) Functional definitions for q-analogues of Eulerian functions and applications. Aequationes mathematicae 85:1-2, pages 69-110.
Crossref
István Mező. (2013) A q -Raabe formula and an integral of the fourth Jacobi theta function. Journal of Number Theory 133:2, pages 692-704.
Crossref
Jian-Ping Fang. (2013) Note on a -Contour Integral Formula of Gasper-Rahman. Abstract and Applied Analysis 2013, pages 1-6.
Crossref
Hamza Menken & Adviye K?r?k??. (2013) Some Properties of the -Extension of the -Adic Gamma Function. Abstract and Applied Analysis 2013, pages 1-4.
Crossref
Ahmed Salem. (2012) A completely monotonic function involving -gamma and -digamma functions . Journal of Approximation Theory 164:7, pages 971-980.
Crossref
Farooq Ahmad, Renu Jain & D. K. Jain. (2012) Some Transformation Formulae of Basic Analogue of I-Function. Asian Journal of Mathematics & Statistics 5:4, pages 158-162.
Crossref
. 2012. Zeta and q-Zeta Functions and Associated Series and Integrals. Zeta and q-Zeta Functions and Associated Series and Integrals 603 657 .
Mahmoud H. Annaby & Zeinab S. MansourMahmoud H. Annaby & Zeinab S. Mansour. 2012. q -Fractional Calculus and Equations. q -Fractional Calculus and Equations 271 293 .
Mahmoud H. Annaby & Zeinab S. MansourMahmoud H. Annaby & Zeinab S. Mansour. 2012. q -Fractional Calculus and Equations. q -Fractional Calculus and Equations 223 270 .
Mahmoud H. Annaby & Zeinab S. MansourMahmoud H. Annaby & Zeinab S. Mansour. 2012. q -Fractional Calculus and Equations. q -Fractional Calculus and Equations 201 222 .
Mahmoud H. Annaby & Zeinab S. MansourMahmoud H. Annaby & Zeinab S. Mansour. 2012. q -Fractional Calculus and Equations. q -Fractional Calculus and Equations 175 199 .
Mahmoud H. Annaby & Zeinab S. MansourMahmoud H. Annaby & Zeinab S. Mansour. 2012. q -Fractional Calculus and Equations. q -Fractional Calculus and Equations 147 173 .
Mahmoud H. Annaby & Zeinab S. MansourMahmoud H. Annaby & Zeinab S. Mansour. 2012. q -Fractional Calculus and Equations. q -Fractional Calculus and Equations 107 146 .
Mahmoud H. Annaby & Zeinab S. MansourMahmoud H. Annaby & Zeinab S. Mansour. 2012. q -Fractional Calculus and Equations. q -Fractional Calculus and Equations 73 105 .
Mahmoud H. Annaby & Zeinab S. MansourMahmoud H. Annaby & Zeinab S. Mansour. 2012. q -Fractional Calculus and Equations. q -Fractional Calculus and Equations 1 39 .
Thomas ErnstThomas Ernst. 2012. A Comprehensive Treatment of q-Calculus. A Comprehensive Treatment of q-Calculus 279 307 .
Thomas ErnstThomas Ernst. 2012. A Comprehensive Treatment of q-Calculus. A Comprehensive Treatment of q-Calculus 195 240 .
D. D. Somashekara, K. Narasimha Murthy & S. L. Shalini. (2011) On a New Summation Formula for Basic Bilateral Hypergeometric Series and Its Applications. International Journal of Mathematics and Mathematical Sciences 2011, pages 1-7.
Crossref
Toufik Mansour. (2009) Identities for Sums of a q-Analogue of Polylogarithm Functions. Letters in Mathematical Physics 87:1-2, pages 1-18.
Crossref
T. Kim & Y. Simsek. (2008) Analytic continuation of the multiple Daehee q-l-functions associated with Daehee numbers. Russian Journal of Mathematical Physics 15:1, pages 58-65.
Crossref
Kevin W. J. Kadell. (2007) Telescoping partial fractions decompositions, the little q-Jacobi functions of complex order, and the nonterminating q-Saalschütz sum. The Ramanujan Journal 13:1-3, pages 449-469.
Crossref
Horst Alzer & Arcadii Z. Grinshpan. (2007) Inequalities for the gamma and q-gamma functions. Journal of Approximation Theory 144:1, pages 67-83.
Crossref
Yoshinori YAMASAKI. (2006) On $q$-Analogues of the Barnes Multiple Zeta Functions. Tokyo Journal of Mathematics 29:2.
Crossref
Geoffrey B. Campbell. (2006) An Euler product transform applied to q-series. The Ramanujan Journal 12:2, pages 267-293.
Crossref
Changgui Zhang. 2005. Theory and Applications of Special Functions. Theory and Applications of Special Functions 479 491 .
Kevin W. J. Kadell. 2005. Theory and Applications of Special Functions. Theory and Applications of Special Functions 301 338 .
A N F Aleixo & A B Balantekin. (2004) An algebraic construction of generalized coherent states for shape-invariant potentials. Journal of Physics A: Mathematical and General 37:35, pages 8513-8528.
Crossref
Nobushige Kurokawa & Masato Wakayama. (2003) On 𝑞-analogues of the Euler constant and Lerch’s limit formula. Proceedings of the American Mathematical Society 132:4, pages 935-943.
Crossref
Jean Bertoin, Philippe Biane & Marc Yor. 2004. Seminar on Stochastic Analysis, Random Fields and Applications IV. Seminar on Stochastic Analysis, Random Fields and Applications IV 45 56 .
J. Sándor & B. CrsticiJ. Sándor & B. Crstici. 2004. Handbook of Number Theory II. Handbook of Number Theory II 459 618 .
Changgui Zhang. (2003) Sur les fonctions q-Bessel de Jackson. Journal of Approximation Theory 122:2, pages 208-223.
Crossref
Renato ?lvarez-Nodarse, M.K. Atakishiyeva & N.M. Atakishiyev. (2003) Mellin transforms for some families of q-polynomials. Journal of Computational and Applied Mathematics 153:1-2, pages 9-18.
Crossref
Hailiang Zhang & Wu Zhang. (2002) Maximum principles and bounds in a class of fourth-order uniformly elliptic equations. Journal of Physics A: Mathematical and General 35:43, pages 9245-9250.
Crossref
A N F Aleixo, A B Balantekin & M A C ndido Ribeiro. (2002) Completeness of coherent states associated with self-similar potentials and Ramanujan s integral extension of the beta function. Journal of Physics A: Mathematical and General 35:43, pages 9063-9069.
Crossref
M.Aslam Chaudhry & S.M. Zubair. (2002) Extended incomplete gamma functions with applications. Journal of Mathematical Analysis and Applications 274:2, pages 725-745.
Crossref
Michitomo Nishizawa. (2002) Multiple Gamma Function, Its $q$- and Elliptic Analogue. Rocky Mountain Journal of Mathematics 32:2.
Crossref
Вадим Валентинович Зудилин & Wadim Valentinovich Zudilin. (2002) О диофантовых задачах для $q$-дзета-значенийDiophantine Problems for $q$-Zeta Values. Математические заметки Matematicheskie Zametki 72:6, pages 936-940.
Crossref
D. S. Lubinsky. 2003. Advanced Problems in Constructive Approximation. Advanced Problems in Constructive Approximation 109 128 .
Horst Alzer. (2001) Sharp Bounds for the Ratio ofq - Gamma Functions. Mathematische Nachrichten 222:1, pages 5-14.
Crossref
Horst Alzer. (2000) Characterizations of the ratio of gamma and q-Gamma functions. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 70:1, pages 165-174.
Crossref
Giovanni Felder & Alexander Varchenko. (2000) The Elliptic Gamma Function and SL(3,?Z)?Z3. Advances in Mathematics 156:1, pages 44-76.
Crossref
Nguyen Xuan Thao. (2000) A basis analog of theH-function of several variables. Mathematical Notes 67:5, pages 625-630.
Crossref
????? ???? ???? Nguyen Xuan Thao. (2000) ???????? ?????? $H$-??????? ?????? ??????????A basis analog of the $H$-function of several variables. ?????????????? ??????? Matematicheskie Zametki 67:5, pages 738-744.
Crossref
Hirofumi Tsumura. (1999) A note on $q$-analogues of Dirichlet series. Proceedings of the Japan Academy, Series A, Mathematical Sciences 75:3.
Crossref
Martin E. Muldoon. 1998. Recent Progress in Inequalities. Recent Progress in Inequalities 309 323 .
Roger Webster. (1997) Log-Convex Solutions to the Functional Equationf(x+1)=g(x)f(x): ?-Type Functions. Journal of Mathematical Analysis and Applications 209:2, pages 605-623.
Crossref
Charles A Nelson & Michael G Gartley. (1996) On the two q -analogue logarithmic functions: . Journal of Physics A: Mathematical and General 29:24, pages 8099-8115.
Crossref
C. Krattenthaler & H.M. Srivastava. (1996) Summations for basic hypergeometric series involving a q-analogue of the digamma function. Computers & Mathematics with Applications 32:3, pages 73-91.
Crossref
Michitomo Nishizawa. (1996) On a q-analogue of the multiple gamma functions. Letters in Mathematical Physics 37:2, pages 201-209.
Crossref
M. Aslam Chaudhry & S. M. Zubair. (2009) On an extension of generalized incomplete Gamma functions with applications. The Journal of the Australian Mathematical Society. Series B. Applied Mathematics 37:3, pages 392-405.
Crossref
M. Aslam Chaudhry & S.M. Zubair. (1995) On the decomposition of generalized incomplete gamma functions with applications to Fourier transforms. Journal of Computational and Applied Mathematics 59:3, pages 253-284.
Crossref
T Prellberg. (1995) Uniform q-series asymptotics for staircase polygons. Journal of Physics A: Mathematical and General 28:5, pages 1289-1304.
Crossref
A. F. Kharshiladze, D. V. Alekseevskiĭ, N. N. Vorob’ev, V. A. Trenogin, A. Z. Petrov, D. D. Sokolov, E. A. Nisnevich, A. I. Skopin, S. P. Demushkin, V. P. Platonov, O. A. Ivanova, L. A. Skornyakov, I. V. Dolgachev, M. M. Postnikov, K. I. Beĭdar, A. V. Mikhalev, B. A. Sevast’yanov, E. B. Yanovskaya, V. K. Domanskiĭ, A. N. Lyapunov, A. S. Mikhaĭlova, F. I. Ereshko, I. A. Vatel’, O. V. Sarmanov, A. V. Prokhorov, Yu. V. Prokhorov, L. P. Kuptsov, A. A. Dezin, Yu. I. Shokin, N. N. Yanenko, L. N. Sretenskiĭ, V. M. Tikhomirov, V. D. Kukin, Yu. D. Burago, D. P. Zhelobenko, M. K. Samarin, S. N. Chernikov, B. M. Bredikhin, V. V. Rumyantsev, N. S. Bakhvalov, V. P. Motornyĭ, S. A. Stepanov, L. N. Shevrin, A. P. Prudnikov, E. D. Solomentsev, R. L. Dobrushin, V. V. Prelov, E. V. Shikin, D. V. Anosov, E. A. Gorin, L. N. Bol’shev, N. Kh. Rozov, V. E. Plisko, A. V. Arkhangel’skiĭ, V. A. Iskovskikh, B. M. Levitan, A. V. Bitsadze, Yu. B. Rudyak, S. M. Nikol’skiĭ, G. L. Litvinov, V. S. Vladimirov, A. L. Shmel’kin, V. I. Ponomarev, P. K. Suetin, S. G. Kreĭn, V. I. Danilov, M. I. Voĭtsekhovskiĭ, A. Wörz-Busekros, G. E. Mints, Vik. S. Kulikov, A. S. Rapinchuk, V. I. Dmitriev, V. A. Zalgaller, Yu. A. Volkov, Yu. S. Slobodyan, V. F. Eremeev, M. I. Yurkina, V. M. Babich, E. G. Sklyarenko, P. S. Modenov, A. S. Parkhomenko, V. M. Kalinin, N. M. Ostianu, E. G. Poznyak, A. D. Aleksandrov, V. A. Toponogov, S. Z. Shefel’, A. V. Malyshev, E. M. Chirka, I. A. Kvasnikov, P. L. Ul’yanov, K. P. Latyshev, I. A. Shishmarev, V. L. Popov, G. V. Kuz’mina, E. N. Kuz’min, A. S. Fedenko, E. G. Goluzina, V. N. Grishin, A. A. Karatsuba, A. G. Dragalin, M. S. Nikulin, V. T. Bazylev, A. M. Nakhushev, A. K. Mitropol’skiĭ, A. V. Gladkiĭ, V. P. Kozyrev, V. B. Alekseev, A. A. Sapozhenko, A. A. Konyushkov, N. M. Nagornyĭ, A. L. Onishchik, N. V. Mitskevich, A. A. Bukhshtab, V. N. Nechaev, A. K. Gushchin, Sh. A. Alimov, V. A. Il’in, V. N. Plechko, L. D. Kudryavtsev, M. Sh. Tsalenko, I. A. Aleksandrov, P. M. Tamrazov, M. I. Kargapolov, Yu. I. Merzlyakov, A. A. Bovdi, A. I. Shtern, E. B. Vinberg, I. P. Egorov, V. M. Kopytov, V. D. Belousov & A. F. Leont’ev. 1995. Encyclopaedia of Mathematics. Encyclopaedia of Mathematics 728 911 .
Detlef Gronau & Janusz Matkowski. (2013) Geometrically Convex Solutions of Certain Difference Equations and Generalized Bohr-Mollerup Type Theorems. Results in Mathematics 26:3-4, pages 290-297.
Crossref
M.Aslam Chaudhry & S.M. Zubair. (1994) Generalized incomplete gamma functions with applications. Journal of Computational and Applied Mathematics 55:1, pages 99-124.
Crossref
Hirofumi Tsumura. (2016) On a q -analogue of the log- Γ -function . Nagoya Mathematical Journal 134, pages 57-64.
Crossref
Taichiro Takagi. (1994) A generalized simple random walk in one dimension related to the Gaussian polynomials. Communications in Mathematical Physics 162:2, pages 261-271.
Crossref
Peter G. O. Freund & Anton V. Zabrodin. (1993) A hierarchic array of integrable models. Journal of Mathematical Physics 34:12, pages 5832-5842.
Crossref
Gang Li & Kai-Tai Fang. (1992) The Ramanujan'sq-extension for the exponential function and statistical distributions. Acta Mathematicae Applicatae Sinica 8:3, pages 264-280.
Crossref
V.K Jain & H.M Srivastava. (1992) New results involving a certain class of q-orthogonal polynomials. Journal of Mathematical Analysis and Applications 166:2, pages 331-344.
Crossref
Roelof Koekoek. (1992) Generalizations of a q-analogue of Laguerre polynomials. Journal of Approximation Theory 69:1, pages 55-83.
Crossref
S. C. Milne. 1992. Progress in Approximation Theory. Progress in Approximation Theory 51 77 .
Mizan Rahman. (1991) Biorthogonality of a System of Rational Functions with Respect to a Positive Measure on $[ - 1,1]$. SIAM Journal on Mathematical Analysis 22:5, pages 1430-1441.
Crossref
R W Gray & C A Nelson. (1990) A completeness relation for the q-analogue coherent states by q-integration. Journal of Physics A: Mathematical and General 23:18, pages L945-L950.
Crossref
Tom H Koornwinder. (1990) Jacobi functions as limit cases of q-ultraspherical polynomials. Journal of Mathematical Analysis and Applications 148:1, pages 44-54.
Crossref
Lutz G. Lucht. (1990) Mittelwertungleichungen für Lösungen gewisser Differenzengleichungen. Aequationes Mathematicae 39:2-3, pages 204-209.
Crossref
Mizan Rahman. 1990. Orthogonal Polynomials. Orthogonal Polynomials 319 344 .
A. P. Golub. (1989) One type of generalized moment representations. Ukrainian Mathematical Journal 41:11, pages 1246-1251.
Crossref
Junya Satoh. (1989) q-Analogue of Riemann's ζ-function and q-Euler numbers. Journal of Number Theory 31:3, pages 346-362.
Crossref
H. M. Srivastava. (1989) Sums of a certain class of $q$-series. Proceedings of the Japan Academy, Series A, Mathematical Sciences 65:1.
Crossref
Richard Askey. 1989. Number Theory, Madras 1987. Number Theory, Madras 1987 84 121 .
M. Hazewinkel. 1989. Encyclopaedia of Mathematics. Encyclopaedia of Mathematics 160 343 .
Laurent Habsieger. (1988) Une q -Int?grale de Selberg et Askey . SIAM Journal on Mathematical Analysis 19:6, pages 1475-1489.
Crossref
S.C Milne. (1988) A q-analog of the Gauss summation theorem for hypergeometric series in U(n). Advances in Mathematics 72:1, pages 59-131.
Crossref
Kevin W. J. Kadell. (1988) A Proof of Askey?s Conjectured q -Analogue of Selberg?s Integral and a Conjecture of Morris . SIAM Journal on Mathematical Analysis 19:4, pages 969-986.
Crossref
Kevin W. J. Kadell. (1988) A Proof of Some q -Analogues of Selberg?s Integral for $k=1$ . SIAM Journal on Mathematical Analysis 19:4, pages 944-968.
Crossref
Kevin W. J. Kadell. (1988) The 𝑞-Selberg polynomials for 𝑛=2. Transactions of the American Mathematical Society 310:2, pages 535-553.
Crossref
K. G. Ramanathan. (1987) Hypergeometric series and continued fractions. Proceedings of the Indian Academy of Sciences - Section A 97:1-3, pages 277-296.
Crossref
Kevin W. J. Kadell. (1987) A Probabilistic Proof of Ramanujan?s ${}_1 \psi _1 $ Sum. SIAM Journal on Mathematical Analysis 18:6, pages 1539-1548.
Crossref
Mourad E.H. Ismail, Dennis Stanton & G?rard Viennot. (1987) The Combinatorics of q-Hermite polynomials and the Askey?Wilson Integral. European Journal of Combinatorics 8:4, pages 379-392.
Crossref
S. K. Suslov. (1987) Classical orthogonal polynomials of a discrete variable continuous orthogonality relation. Letters in Mathematical Physics 14:1, pages 77-88.
Crossref
S.C Milne. (1987) Basic hypergeometric series very well-poised in U(n). Journal of Mathematical Analysis and Applications 122:1, pages 223-256.
Crossref
John Greene & Dennis Stanton. (1986) The triplication formula for Gauss sums. Aequationes Mathematicae 30:1, pages 134-141.
Crossref
Mourad E.H Ismail, Lee Lorch & Martin E Muldoon. (1986) Completely monotonic functions associated with the gamma function and its q-analogues. Journal of Mathematical Analysis and Applications 116:1, pages 1-9.
Crossref
A. F. Nikiforov & S. K. Suslov. (1986) Classical orthogonal polynomials of a discrete variable on nonuniform lattices. Letters in Mathematical Physics 11:1, pages 27-34.
Crossref
P.I. Pastro. (1985) Orthogonal polynomials and some q-beta integrals of Ramanujan. Journal of Mathematical Analysis and Applications 112:2, pages 517-540.
Crossref
Hans-Heinrich Kairies. (1985) Einbettungen von log Γ durch Scharen spezieller Krull-Normallösungen. Aequationes Mathematicae 29:1, pages 28-35.
Crossref
B. Nassrallah & Mizan Rahman. (1985) Projection Formulas, a Reproducing Kernel and a Generating Function for q -Wilson Polynomials . SIAM Journal on Mathematical Analysis 16:1, pages 186-197.
Crossref
Waleed Al-Salam, W. R. Allaway & Richard Askey. (1984) Sieved ultraspherical polynomials. Transactions of the American Mathematical Society 284:1, pages 39-55.
Crossref
Mizan Rahman. (1984) A simple evaluation of Askey and Wilson’s 𝑞-beta integral. Proceedings of the American Mathematical Society 92:3, pages 413-417.
Crossref
Hans-Heinrich Kairies. 1984. General Inequalities 4. General Inequalities 4 257 267 .
H. -H. Kairies & M. E. Muldoon. (1982) Some characterizations ofq-factorial functions. Aequationes Mathematicae 24:1, pages 319-319.
Crossref
H. -H. Kairies & M. E. Muldoon. (1982) Some characterizations ofq-factorial functions. Aequationes Mathematicae 25:1, pages 67-76.
Crossref
Mourad E.H Ismail. (1982) The zeros of basic Bessel functions, the functions Jv + ax(x), and associated orthogonal polynomials. Journal of Mathematical Analysis and Applications 86:1, pages 1-19.
Crossref
Neal Koblitz. (1982) 𝑞-extension of the 𝑝-adic gamma function. II. Transactions of the American Mathematical Society 273:1, pages 111-129.
Crossref
Richard Askey. 1982. Gabor Szegö: Collected Papers. Gabor Szegö: Collected Papers 433 439 .
Daniel S Moak. (1981) The q-analogue of the Laguerre polynomials. Journal of Mathematical Analysis and Applications 81:1, pages 20-47.
Crossref
Richard Askey. (1980) Some Basic Hypergeometric Extensions of Integrals of Selberg and Andrews. SIAM Journal on Mathematical Analysis 11:6, pages 938-951.
Crossref
Neal Koblitz. (1980) 𝑞-extension of the 𝑝-adic gamma function. Transactions of the American Mathematical Society 260:2, pages 449-457.
Crossref
Richard Askey & Mourad E. H. Ismail. (1979) The very well poised ₆𝜓₆. Proceedings of the American Mathematical Society 77:2, pages 218-222.
Crossref

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Order Reprints Request Corporate Permissions

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

Request Academic Permissions

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.

Related research

People also read lists articles that other readers of this article have read.

Recommended articles lists articles that we recommend and is powered by our AI driven recommendation engine.

Cited by lists all citing articles based on Crossref citations.
Articles with the Crossref icon will open in a new tab.