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A conforming-nonconforming mixed immersed finite element method for unsteady Stokes equations with moving interfaces
Telescoping method, summation formulas, and inversion pairs
School of Mathematics, Tianjin University, Tianjin 300350, China |
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.
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. |
[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. |
[3] |
D. M. Bressoud,
A matrix inverse, Proc. Amer. Math. Soc., 88 (1983), 446-448.
doi: 10.1090/S0002-9939-1983-0699411-9. |
[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. |
[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. |
[6] |
G. Gasper and M. Rahman, Basic Hypergeometric Series, Second Ed., Cambridge University Press, Cambridge, 2004.
doi: 10.1017/CBO9780511526251. |
[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. |
[8] |
M. Karr,
Summation in finite terms, J. Assoc. Comput. Mach., 28 (1981), 305-350.
doi: 10.1145/322248.322255. |
[9] |
C. Krattenthaler,
A new matrix inverse, Proc. Amer. Math. Soc., 124 (1996), 47-59.
doi: 10.1090/S0002-9939-96-03042-0. |
[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. |
[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. 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. |
[15] |
D. Zeilberger,
The method of creative telescoping, J. Symbolic Comput., 11 (1991), 195-204.
doi: 10.1016/S0747-7171(08)80044-2. |
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. |
[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. |
[3] |
D. M. Bressoud,
A matrix inverse, Proc. Amer. Math. Soc., 88 (1983), 446-448.
doi: 10.1090/S0002-9939-1983-0699411-9. |
[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. |
[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. |
[6] |
G. Gasper and M. Rahman, Basic Hypergeometric Series, Second Ed., Cambridge University Press, Cambridge, 2004.
doi: 10.1017/CBO9780511526251. |
[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. |
[8] |
M. Karr,
Summation in finite terms, J. Assoc. Comput. Mach., 28 (1981), 305-350.
doi: 10.1145/322248.322255. |
[9] |
C. Krattenthaler,
A new matrix inverse, Proc. Amer. Math. Soc., 124 (1996), 47-59.
doi: 10.1090/S0002-9939-96-03042-0. |
[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. |
[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. 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. |
[15] |
D. Zeilberger,
The method of creative telescoping, J. Symbolic Comput., 11 (1991), 195-204.
doi: 10.1016/S0747-7171(08)80044-2. |
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