A MULTIPLE q-EXPONENTIAL DIFFERENTIAL OPERATIONAL IDENTITY*

Abstract

References

Metrics

Comments

Recommended 10

doi:10.1007/s10473-023-0608-3

Abstract: Using Hartogs' fundamental theorem for analytic functions in several complex variables and $q$ -partial differential equations, we establish a multiple $q$ -exponential differential formula for analytic functions in several variables. With this identity, we give new proofs of a variety of important classical formulas including Bailey's $_6\psi_6$ series summation formula and the Atakishiyev integral. A new transformation formula for a double $q$ -series with several interesting special cases is given. A new transformation formula for a $_3\psi_3$ series is proved.

Key words: $q$ -hypergeometric series, $q$ -exponential differential operator, Bailey's $_6\psi_6$ summation, double $q$ -hypergeometric series, $q$ -partial differential equation

CLC Number:

05A30

Zhiguo LIU. A MULTIPLE q-EXPONENTIAL DIFFERENTIAL OPERATIONAL IDENTITY^*[J].Acta mathematica scientia,Series B, 2023, 43(6): 2449-2470.

Trendmd

[1] Andrews G E.

q

$q$ -Orthogonal polynomials, Rogers-Ramanujan identities and mock theta functions. Proc Steklov Inst Math, 2012, 276: 21-32
[2] Askey R.Beta integrals and

q

$q$ -extensions//Proceedings of the Ramanujan Centennial International Conference. Annamalainagar, 15-18 December, 1987: 85-102
[3] Aslan H, Ismail M E H. A

q

$q$ -translation to Liu's calculus. Ann Comb, 2019, 23: 465-488
[4] Atakishiyev N M. On the Askey-Wilson

q

$q$ -beta integral. Teoret Mat Fiz, 1994, 99: 155-159
[5] Bailey W N. On the basic bilateral hypergeometric series

_{2} ψ_{2}

$_2\psi_2$ . Quart J Math (Oxford), 1950, 1: 194-198
[6] Bhatnagar G, Rai S. Expansion formulas for multiple basic hypergeometric series over root systems. Adv Appl Math, 2022, 137: 102329
[7] Carlitz L. Some polynomials related to theta functions. Ann Mat Pure Appl, 1955, 41: 359-373
[8] Chen D, Wang L, Representations of mock theta functions. Adv Math, 2020, 365: 107037
[9] Chen W Y C, Liu Z G. Parameter augmenting for basic hypergeometric series I//Sagan B E, Stanley R P. Mathematical Essays in Honor of Gian-Carlo Rota. Basel: Birkuser, 1998: 111-129
[10] Cui S P, Gu N S S. Some new mock theta functions. Adv Appl Math, 2021, 131: 102267
[11] Gasper G, Rahman M. Basic Hypergeometric Series.2nd ed. Cambridge: Cambridge Univ Press, 2004
[12] Jackson F H. On

q

$q$ -functions and a certain difference operator. Trans Roy Soc Edin, 1908, 46: 253-281
[13] Liu Z G. An expansion formula for

q

$q$ -series and applications. Ramanujan J, 2002, 6: 429-447
[14] Liu Z G. Some operator identities and

q

$q$ -series transformation formulas. Discret Math, 2003, 265: 119-139
[15] Liu Z G. Two

q

$q$ -difference equations and

q

$q$ -operator identities. J Difference Equ Appl, 2010, 16: 1293-1307
[16] Liu Z G. An extension of the non-terminating

_{6} ϕ_{5}

${}_6\phi_5$ summation and the Askey-Wilson polynomials. J Difference Equ Appl, 2011, 17: 1401-1411
[17] Liu Z G.On the

q

$q$ -partial differential equations and

q

$q$ -series//The Legacy of Srinivasa Ramanujan Ramanujan Math Soc Lect Notes Ser, Vol 20. Mysore: Ramanujan Math Soc, 2013: 213-250
[18] Liu Z G. A

q

$q$ -series expansion formula and the Askey-Wilson polynomials. Ramanujan J, 2013, 30: 193-210
[19] Liu Z G.On the

q

$q$ -derivative and

q

$q$ -series expansions. Int J Number Theory, 2013, 9: 2069-2089
[20] Liu Z G. A

q

$q$ -extension of a partial differential equation and the Hahn polynomials. Ramanujan J, 2015, 38: 481-501
[21] Liu Z G. On a reduction formula for a kind of double

q

$q$ -integrals. Symmetry, 2016, 8(6): 44
[22] Liu Z G. Extensions of Ramanujan's reciprocity theorem and the Andrews-Askey integral. J Math Anal Appl, 2016, 443: 1110-1129
[23] Liu Z G.On a system of

q

$q$ -partial differential equations with applications to

q

$q$ -series//Analytic Number Theory, Modular Forms and

q

$q$ -hypergeometric Series. Springer Proc Math Stat, 221. Cham: Springer, 2017: 445-461
[24] Liu Z G. A

q

$q$ -operational equation and the Rogers-Szegö polynomials. Sci China Math, 2023. https://doi.org/10.1007/s11425-021-1999-2
[25] Liu Z G.A multiple

q

$q$ -translation formula and its implications. Acta Math Sin-English Ser, DOI:10.1007/S10114-023-2237-0
[26] Malgrange B.Lectures on Functions of Several Complex Variables. Berlin: Springer, 1984
[27] Range R M. Complex analysis: A brief tour into higher dimensions. Amer Math Monthly, 2003, 110: 89-108
[28] Rogers L J. On a three-fold symmetry in the elements of Heine's series. Proc London Math Soc, 1893, 24: 171-179
[29] Schendel L. Zur theorie der functionen. J Reine Angew Math, 1878, 84: 80-84
[30] Szegö G. Ein Betrag zur Theorie der Thetafunktionen. Sitz Preuss Akad Wiss Phys Math, 1926, 19: 242-252
[31] Taylor J.Several Complex Variables with Connections to Algebraic Geometry and Lie Groups. Graduate Studies in Mathematics, Vol 46. Providence: Amer Math Soc, 2002
[32] Wang C, Chern S. Some

q

$q$ -transformation formulas and Hecke-type identities. Int J Number Theory, 2019, 15: 1349-1367
[33] Wang J, Ma X. An expansion formula of basic hypergeometric series via the

(1 - x y, y - x)

$(1-xy, y-x)$ -inversion and its applications. arXiv:1301.3582, 2021
[34] Zhang Helen W J. Further generalizations of some Hecke-type identities. Int J Number Theory, 2022, 18: 361-387
[35] Zhang W J. Further extensions of some truncated Hecke type identities. Acta Mathematica Scientia, 2022, 42B(1): 73-90
[36] Zhang Y, Zhang W. Hecke-Rogers type series representations for infinite products. Ramanujan J, 2022, 58: 889-903

Viewed

Full text

	From	local

	Times	6
	Rate	100%

Abstract

Just accepted	Online first	Issue

0	0	37

	From	Others

	Times	37
	Rate	100%

Cited

Web of Science	Crossref	ScienceDirect	Search for Citations in Google Scholar >>


This page requires you have already subscribed to WoS.

Shared

Discussed

Just accepted

Online first

Just accepted

Online first

[1]	Sitan Lin. OBSTACLE PROBLEMS ON $RCD(K, N)$ METRIC MEASURE SPACES^*[J]. Acta mathematica scientia,Series B, 2023, 43(5): 1925 -1944 .
[2]	Yirang Yuan, Changfeng Li, Tongjun Sun, Qing Yang. A MIXED FINITE ELEMENT AND CHARACTERISTIC MIXED FINITE ELEMENT FOR INCOMPRESSIBLE MISCIBLE DARCY-FORCHHEIMER DISPLACEMENT AND NUMERICAL ANALYSIS^*[J]. Acta mathematica scientia,Series B, 2023, 43(5): 2026 -2042 .
[3]	Léo Glangetas, Van-Sang Ngo, El Mehdi Said. HYDROSTATIC LIMIT OF THE NAVIER-STOKES-ALPHA MODEL^*[J]. Acta mathematica scientia,Series B, 2023, 43(5): 1945 -1980 .
[4]	Yanyan han, Huoxiong wu. THE FRACTIONAL TYPE MARCINKIEWICZ INTEGRALS AND COMMUTATORS ON WEIGHTED HARDY SPACES^*[J]. Acta mathematica scientia,Series B, 2023, 43(5): 1981 -1996 .
[5]	Shanli YE, Guanghao FENG. A DERIVATIVE-HILBERT OPERATOR ACTING ON HARDY SPACES^*[J]. Acta mathematica scientia,Series B, 2023, 43(6): 2398 -2412 .
[6]	Huiyang XU, Cece LI. CONFORMALLY FLAT AFFINE HYPERSURFACES WITH SEMI-PARALLEL CUBIC FORM^*[J]. Acta mathematica scientia,Series B, 2023, 43(6): 2413 -2429 .
[7]	Fenglong Qu, Ruixue Jia, Yanli Cui. INVERSE CONDUCTIVE MEDIUM SCATTERING WITH UNKNOWN BURIED OBJECTS^*[J]. Acta mathematica scientia,Series B, 2023, 43(5): 2005 -2025 .
[8]	Ruinan LI, Xinyu WANG. TRANSPORTATION COST-INFORMATION INEQUALITY FOR A STOCHASTIC HEAT EQUATION DRIVEN BY FRACTIONAL-COLORED NOISE^*[J]. Acta mathematica scientia,Series B, 2023, 43(6): 2519 -2532 .
[9]	Shengchuang CHANG, Ran DUAN. ZERO DISSIPATION LIMIT TO A RAREFACTION WAVE WITH A VACUUM FOR A COMPRESSIBLE, HEAT CONDUCTING REACTING MIXTURE^*[J]. Acta mathematica scientia,Series B, 2023, 43(6): 2533 -2552 .
[10]	Qinghui LIU, Zhiyi Tang. THE HAUSDORFF DIMENSION OF THE SPECTRUM OF A CLASS OF GENERALIZED THUE-MORSE HAMILTONIANS^*[J]. Acta mathematica scientia,Series B, 2023, 43(5): 1997 -2004 .