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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

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