American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Continuous data assimilation applied to a velocity-vorticity formulation of the 2D Navier-Stokes equations
  • ERA Home
  • This Issue
  • Next Article
    Decoupling PDE computation with intrinsic or inertial Robin interface condition
doi: 10.3934/era.2021007

Telescoping method, summation formulas, and inversion pairs

Qing-Hu Hou and  Yarong Wei ,

School of Mathematics, Tianjin University, Tianjin 300350, China

* Corresponding author

Received  May 2020 Revised  November 2020 Published  January 2021

Based on Gosper's algorithm, we present an approach to the telescoping of general sequences. Along this approach, we propose a summation formula and a bibasic extension of Ma's inversion formula. From the formulas, we are able to derive several hypergeometric and elliptic hypergeometric identities.

Keywords: Telescoping method, Gosper's algorithm, inversion pair, basic hypergeometric series, elliptic hypergeometric series.
Mathematics Subject Classification: 33F10, 05A19.
Citation: Qing-Hu Hou, Yarong Wei. Telescoping method, summation formulas, and inversion pairs. Electronic Research Archive, doi: 10.3934/era.2021007
References:
[1]

A. Bauer and M. Petkovšek, Multibasic and mixed hypergeometric Gosper-type algorithms, J. Symbolic Comput., 28 (1999), 711-736.  doi: 10.1006/jsco.1999.0321.  Google Scholar

[2]

G. Bhatnagar and S. C. Milne, Generalized bibasic hypergeometric series and their $$ \rm U $(n)$ extensions, Adv. Math., 131 (1997), 188-252.  doi: 10.1006/aima.1997.1659.  Google Scholar

[3]

D. M. Bressoud, A matrix inverse, Proc. Amer. Math. Soc., 88 (1983), 446-448.  doi: 10.1090/S0002-9939-1983-0699411-9.  Google Scholar

[4]

F. Chyzak, An extension of Zeilberger's fast algorithm to general holonomic functions, Discrete Math., 217 (2000), 115-134.  doi: 10.1016/S0012-365X(99)00259-9.  Google Scholar

[5]

G. Gasper, Summation, transformation, and expansion formulas for bibasic series, Trans. Amer. Math. Soc., 312 (1989), 257-277.  doi: 10.1090/S0002-9947-1989-0953537-0.  Google Scholar

[6]

G. Gasper and M. Rahman, Basic Hypergeometric Series, Second Ed., Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511526251.  Google Scholar

[7]

R. W. Gosper Jr., Decision procedure for indefinite hypergeometric summation, Proc. Natl. Acad. Sci. USA, 75 (1978), 40-42.  doi: 10.1073/pnas.75.1.40.  Google Scholar

[8]

M. Karr, Summation in finite terms, J. Assoc. Comput. Mach., 28 (1981), 305-350.  doi: 10.1145/322248.322255.  Google Scholar

[9]

C. Krattenthaler, A new matrix inverse, Proc. Amer. Math. Soc., 124 (1996), 47-59.  doi: 10.1090/S0002-9939-96-03042-0.  Google Scholar

[10]

X. Ma, The $(f, g)$-inversion formula and its applications: The $(f, g)$-summation formula, Adv. in Appl. Math., 38 (2007), 227-257.  doi: 10.1016/j.aam.2005.06.006.  Google Scholar

[11]

P. Paule and C. Schneider, Towards a symbolic summation theory for unspecified sequences, In: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, 351–390, Texts Monogr. Symbol. Comput., Springer, Cham, 2019.  Google Scholar

[12]

M. Petkovšek, H. S. Wilf and D. Zeilberger, $A = B$, A K Peters, Ltd., Wellesley, MA, 1996.  Google Scholar

[13]

C. Schneider, Symbolic Summation in Difference Fields, Ph. D. thesis, J. Kepler University, 2001. Google Scholar

[14]

S. O. Warnaar, Summation and transformation formulas for elliptic hypergeometric series, Constr. Approx., 18 (2002), 479-502.  doi: 10.1007/s00365-002-0501-6.  Google Scholar

[15]

D. Zeilberger, The method of creative telescoping, J. Symbolic Comput., 11 (1991), 195-204.  doi: 10.1016/S0747-7171(08)80044-2.  Google Scholar

show all references

References:
[1]

A. Bauer and M. Petkovšek, Multibasic and mixed hypergeometric Gosper-type algorithms, J. Symbolic Comput., 28 (1999), 711-736.  doi: 10.1006/jsco.1999.0321.  Google Scholar

[2]

G. Bhatnagar and S. C. Milne, Generalized bibasic hypergeometric series and their $$ \rm U $(n)$ extensions, Adv. Math., 131 (1997), 188-252.  doi: 10.1006/aima.1997.1659.  Google Scholar

[3]

D. M. Bressoud, A matrix inverse, Proc. Amer. Math. Soc., 88 (1983), 446-448.  doi: 10.1090/S0002-9939-1983-0699411-9.  Google Scholar

[4]

F. Chyzak, An extension of Zeilberger's fast algorithm to general holonomic functions, Discrete Math., 217 (2000), 115-134.  doi: 10.1016/S0012-365X(99)00259-9.  Google Scholar

[5]

G. Gasper, Summation, transformation, and expansion formulas for bibasic series, Trans. Amer. Math. Soc., 312 (1989), 257-277.  doi: 10.1090/S0002-9947-1989-0953537-0.  Google Scholar

[6]

G. Gasper and M. Rahman, Basic Hypergeometric Series, Second Ed., Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511526251.  Google Scholar

[7]

R. W. Gosper Jr., Decision procedure for indefinite hypergeometric summation, Proc. Natl. Acad. Sci. USA, 75 (1978), 40-42.  doi: 10.1073/pnas.75.1.40.  Google Scholar

[8]

M. Karr, Summation in finite terms, J. Assoc. Comput. Mach., 28 (1981), 305-350.  doi: 10.1145/322248.322255.  Google Scholar

[9]

C. Krattenthaler, A new matrix inverse, Proc. Amer. Math. Soc., 124 (1996), 47-59.  doi: 10.1090/S0002-9939-96-03042-0.  Google Scholar

[10]

X. Ma, The $(f, g)$-inversion formula and its applications: The $(f, g)$-summation formula, Adv. in Appl. Math., 38 (2007), 227-257.  doi: 10.1016/j.aam.2005.06.006.  Google Scholar

[11]

P. Paule and C. Schneider, Towards a symbolic summation theory for unspecified sequences, In: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, 351–390, Texts Monogr. Symbol. Comput., Springer, Cham, 2019.  Google Scholar

[12]

M. Petkovšek, H. S. Wilf and D. Zeilberger, $A = B$, A K Peters, Ltd., Wellesley, MA, 1996.  Google Scholar

[13]

C. Schneider, Symbolic Summation in Difference Fields, Ph. D. thesis, J. Kepler University, 2001. Google Scholar

[14]

S. O. Warnaar, Summation and transformation formulas for elliptic hypergeometric series, Constr. Approx., 18 (2002), 479-502.  doi: 10.1007/s00365-002-0501-6.  Google Scholar

[15]

D. Zeilberger, The method of creative telescoping, J. Symbolic Comput., 11 (1991), 195-204.  doi: 10.1016/S0747-7171(08)80044-2.  Google Scholar

[1]

Cheng Peng, Zhaohui Tang, Weihua Gui, Qing Chen, Jing He. A bidirectional weighted boundary distance algorithm for time series similarity computation based on optimized sliding window size. Journal of Industrial & Management Optimization, 2021, 17 (1) : 205-220. doi: 10.3934/jimo.2019107
[2]

Editorial Office. Retraction: Xiao-Qian Jiang and Lun-Chuan Zhang, Stock price fluctuation prediction method based on time series analysis. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 915-915. doi: 10.3934/dcdss.2019061
[3]

Kateřina Škardová, Tomáš Oberhuber, Jaroslav Tintěra, Radomír Chabiniok. Signed-distance function based non-rigid registration of image series with varying image intensity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1145-1160. doi: 10.3934/dcdss.2020386
[4]

Lan Luo, Zhe Zhang, Yong Yin. Simulated annealing and genetic algorithm based method for a bi-level seru loading problem with worker assignment in seru production systems. Journal of Industrial & Management Optimization, 2021, 17 (2) : 779-803. doi: 10.3934/jimo.2019134
[5]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056
[6]

Wenya Qi, Padmanabhan Seshaiyer, Junping Wang. A four-field mixed finite element method for Biot's consolidation problems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020127
[7]

Maika Goto, Kazunori Kuwana, Yasuhide Uegata, Shigetoshi Yazaki. A method how to determine parameters arising in a smoldering evolution equation by image segmentation for experiment's movies. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 881-891. doi: 10.3934/dcdss.2020233
[8]

Waixiang Cao, Lueling Jia, Zhimin Zhang. A $ C^1 $ Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 81-105. doi: 10.3934/dcdsb.2020327
[9]

Zhiyan Ding, Qin Li, Jianfeng Lu. Ensemble Kalman Inversion for nonlinear problems: Weights, consistency, and variance bounds. Foundations of Data Science, 2020  doi: 10.3934/fods.2020018
[10]

Wolfgang Riedl, Robert Baier, Matthias Gerdts. Optimization-based subdivision algorithm for reachable sets. Journal of Computational Dynamics, 2021, 8 (1) : 99-130. doi: 10.3934/jcd.2021005
[11]

Hong Fu, Mingwu Liu, Bo Chen. Supplier's investment in manufacturer's quality improvement with equity holding. Journal of Industrial & Management Optimization, 2021, 17 (2) : 649-668. doi: 10.3934/jimo.2019127
[12]

Skyler Simmons. Stability of broucke's isosceles orbit. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021015
[13]

Neil S. Trudinger, Xu-Jia Wang. Quasilinear elliptic equations with signed measure. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 477-494. doi: 10.3934/dcds.2009.23.477
[14]

François Ledrappier. Three problems solved by Sébastien Gouëzel. Journal of Modern Dynamics, 2020, 16: 373-387. doi: 10.3934/jmd.2020015
[15]

Ugo Bessi. Another point of view on Kusuoka's measure. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020404
[16]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272
[17]

Lucio Damascelli, Filomena Pacella. Sectional symmetry of solutions of elliptic systems in cylindrical domains. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3305-3325. doi: 10.3934/dcds.2020045
[18]

Ole Løseth Elvetun, Bjørn Fredrik Nielsen. A regularization operator for source identification for elliptic PDEs. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021006
[19]

Pierre Baras. A generalization of a criterion for the existence of solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 465-504. doi: 10.3934/dcdss.2020439
[20]

Nahed Naceur, Nour Eddine Alaa, Moez Khenissi, Jean R. Roche. Theoretical and numerical analysis of a class of quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 723-743. doi: 10.3934/dcdss.2020354

 Impact Factor: 0.263

Tools

Metrics

  • PDF downloads (11)
  • HTML views (25)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences