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.
Citation: Qing-Hu Hou, Yarong Wei. Telescoping method, summation formulas, and inversion pairs[J]. Electronic Research Archive, 2021, 29(4): 2657-2671. doi: 10.3934/era.2021007
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.
[1] |
Multibasic and mixed hypergeometric Gosper-type algorithms. J. Symbolic Comput. (1999) 28: 711-736. ![]() |
[2] |
Generalized bibasic hypergeometric series and their $![]() |
[3] |
A matrix inverse. Proc. Amer. Math. Soc. (1983) 88: 446-448. ![]() |
[4] |
An extension of Zeilberger's fast algorithm to general holonomic functions. Discrete Math. (2000) 217: 115-134. ![]() |
[5] |
Summation, transformation, and expansion formulas for bibasic series. Trans. Amer. Math. Soc. (1989) 312: 257-277. ![]() |
[6] |
G. Gasper and M. Rahman, Basic Hypergeometric Series, Second Ed., Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511526251
![]() |
[7] |
Decision procedure for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. USA (1978) 75: 40-42. ![]() |
[8] |
Summation in finite terms. J. Assoc. Comput. Mach. (1981) 28: 305-350. ![]() |
[9] |
A new matrix inverse. Proc. Amer. Math. Soc. (1996) 124: 47-59. ![]() |
[10] |
The $(f, g)$-inversion formula and its applications: The $(f, g)$-summation formula. Adv. in Appl. Math. (2007) 38: 227-257. ![]() |
[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. |
[12] | M. Petkovšek, H. S. Wilf and D. Zeilberger, $A = B$, A K Peters, Ltd., Wellesley, MA, 1996. |
[13] | C. Schneider, Symbolic Summation in Difference Fields, Ph. D. thesis, J. Kepler University, 2001. |
[14] |
Summation and transformation formulas for elliptic hypergeometric series. Constr. Approx. (2002) 18: 479-502. ![]() |
[15] |
The method of creative telescoping. J. Symbolic Comput. (1991) 11: 195-204. ![]() |