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September  2021, 29(4): 2657-2671. 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  September 2021 Early access  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, 2021, 29 (4) : 2657-2671. 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

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