Loading…
A telescoping method for double summations
We present a method to prove hypergeometric double summation identities. Given a hypergeometric term F ( n , i , j ) , we aim to find a difference operator L = a 0 ( n ) N 0 + a 1 ( n ) N 1 + ⋯ + a r ( n ) N r and rational functions R 1 ( n , i , j ) , R 2 ( n , i , j ) such that LF = Δ i ( R 1 F )...
Saved in:
| Published in: | Journal of computational and applied mathematics 2006-11, Vol.196 (2), p.553-566 |
|---|---|
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| cited_by | cdi_FETCH-LOGICAL-c401t-9a4f62758973655e3f2a9593c778a01ece125472426a15b3d4ae8e830b12a2513 |
|---|---|
| cites | cdi_FETCH-LOGICAL-c401t-9a4f62758973655e3f2a9593c778a01ece125472426a15b3d4ae8e830b12a2513 |
| container_end_page | 566 |
| container_issue | 2 |
| container_start_page | 553 |
| container_title | Journal of computational and applied mathematics |
| container_volume | 196 |
| creator | Chen, William Y.C. Hou, Qing-Hu Mu, Yan-Ping |
| description | We present a method to prove hypergeometric double summation identities. Given a hypergeometric term
F
(
n
,
i
,
j
)
, we aim to find a difference operator
L
=
a
0
(
n
)
N
0
+
a
1
(
n
)
N
1
+
⋯
+
a
r
(
n
)
N
r
and rational functions
R
1
(
n
,
i
,
j
)
,
R
2
(
n
,
i
,
j
)
such that
LF
=
Δ
i
(
R
1
F
)
+
Δ
j
(
R
2
F
)
. Based on simple divisibility considerations, we show that the denominators of
R
1
and
R
2
must possess certain factors which can be computed from
F
(
n
,
i
,
j
)
. Using these factors as estimates, we may find the numerators of
R
1
and
R
2
by guessing the upper bounds of the degrees and solving systems of linear equations. Our method is valid for the Andrews–Paule identity, Carlitz's identities, the Apéry–Schmidt–Strehl identity, the Graham–Knuth–Patashnik identity, and the Petkovšek–Wilf–Zeilberger identity. |
| doi_str_mv | 10.1016/j.cam.2005.10.010 |
| format | article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_miscellaneous_28672434</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><els_id>S0377042705006205</els_id><sourcerecordid>28672434</sourcerecordid><originalsourceid>FETCH-LOGICAL-c401t-9a4f62758973655e3f2a9593c778a01ece125472426a15b3d4ae8e830b12a2513</originalsourceid><addsrcrecordid>eNp9kE1LxDAQhoMouK7-AG-96EHomslH0-BpEb9gwYueQzadapa2WZNW8N-bZRe8eRpmeN53Zl5CLoEugEJ1u1k42y8YpTL3Cwr0iMygVroEpepjMqNcqZIKpk7JWUobSmmlQczIzbIYscPkwtYPH0WP42doijbEognTusMiTX1vRx-GdE5OWtslvDjUOXl_fHi7fy5Xr08v98tV6QSFsdRWtBVTstaKV1Iib5nVUnOXD7EU0CEwKRQTrLIg17wRFmusOV0Ds0wCn5Prve82hq8J02h6nxx2nR0wTMmwuspqLjIIe9DFkFLE1myj7238MUDNLhWzMTkVs0tlN8qpZM3VwdwmZ7s22sH59CdUWoOu6szd7TnMn357jCY5j4PDxkd0o2mC_2fLL65ddNc</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>28672434</pqid></control><display><type>article</type><title>A telescoping method for double summations</title><source>Elsevier</source><creator>Chen, William Y.C. ; Hou, Qing-Hu ; Mu, Yan-Ping</creator><creatorcontrib>Chen, William Y.C. ; Hou, Qing-Hu ; Mu, Yan-Ping</creatorcontrib><description>We present a method to prove hypergeometric double summation identities. Given a hypergeometric term
F
(
n
,
i
,
j
)
, we aim to find a difference operator
L
=
a
0
(
n
)
N
0
+
a
1
(
n
)
N
1
+
⋯
+
a
r
(
n
)
N
r
and rational functions
R
1
(
n
,
i
,
j
)
,
R
2
(
n
,
i
,
j
)
such that
LF
=
Δ
i
(
R
1
F
)
+
Δ
j
(
R
2
F
)
. Based on simple divisibility considerations, we show that the denominators of
R
1
and
R
2
must possess certain factors which can be computed from
F
(
n
,
i
,
j
)
. Using these factors as estimates, we may find the numerators of
R
1
and
R
2
by guessing the upper bounds of the degrees and solving systems of linear equations. Our method is valid for the Andrews–Paule identity, Carlitz's identities, the Apéry–Schmidt–Strehl identity, the Graham–Knuth–Patashnik identity, and the Petkovšek–Wilf–Zeilberger identity.</description><identifier>ISSN: 0377-0427</identifier><identifier>EISSN: 1879-1778</identifier><identifier>DOI: 10.1016/j.cam.2005.10.010</identifier><identifier>CODEN: JCAMDI</identifier><language>eng</language><publisher>Amsterdam: Elsevier B.V</publisher><subject>Algorithmics. Computability. Computer arithmetics ; Applied sciences ; Computer science; control theory; systems ; Double summation ; Exact sciences and technology ; Hypergeometric term ; Mathematical analysis ; Mathematics ; Sciences and techniques of general use ; Special functions ; Theoretical computing ; Zeilberger's algorithm</subject><ispartof>Journal of computational and applied mathematics, 2006-11, Vol.196 (2), p.553-566</ispartof><rights>2005 Elsevier B.V.</rights><rights>2006 INIST-CNRS</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c401t-9a4f62758973655e3f2a9593c778a01ece125472426a15b3d4ae8e830b12a2513</citedby><cites>FETCH-LOGICAL-c401t-9a4f62758973655e3f2a9593c778a01ece125472426a15b3d4ae8e830b12a2513</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,776,780,27901,27902</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=17991968$$DView record in Pascal Francis$$Hfree_for_read</backlink></links><search><creatorcontrib>Chen, William Y.C.</creatorcontrib><creatorcontrib>Hou, Qing-Hu</creatorcontrib><creatorcontrib>Mu, Yan-Ping</creatorcontrib><title>A telescoping method for double summations</title><title>Journal of computational and applied mathematics</title><description>We present a method to prove hypergeometric double summation identities. Given a hypergeometric term
F
(
n
,
i
,
j
)
, we aim to find a difference operator
L
=
a
0
(
n
)
N
0
+
a
1
(
n
)
N
1
+
⋯
+
a
r
(
n
)
N
r
and rational functions
R
1
(
n
,
i
,
j
)
,
R
2
(
n
,
i
,
j
)
such that
LF
=
Δ
i
(
R
1
F
)
+
Δ
j
(
R
2
F
)
. Based on simple divisibility considerations, we show that the denominators of
R
1
and
R
2
must possess certain factors which can be computed from
F
(
n
,
i
,
j
)
. Using these factors as estimates, we may find the numerators of
R
1
and
R
2
by guessing the upper bounds of the degrees and solving systems of linear equations. Our method is valid for the Andrews–Paule identity, Carlitz's identities, the Apéry–Schmidt–Strehl identity, the Graham–Knuth–Patashnik identity, and the Petkovšek–Wilf–Zeilberger identity.</description><subject>Algorithmics. Computability. Computer arithmetics</subject><subject>Applied sciences</subject><subject>Computer science; control theory; systems</subject><subject>Double summation</subject><subject>Exact sciences and technology</subject><subject>Hypergeometric term</subject><subject>Mathematical analysis</subject><subject>Mathematics</subject><subject>Sciences and techniques of general use</subject><subject>Special functions</subject><subject>Theoretical computing</subject><subject>Zeilberger's algorithm</subject><issn>0377-0427</issn><issn>1879-1778</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2006</creationdate><recordtype>article</recordtype><recordid>eNp9kE1LxDAQhoMouK7-AG-96EHomslH0-BpEb9gwYueQzadapa2WZNW8N-bZRe8eRpmeN53Zl5CLoEugEJ1u1k42y8YpTL3Cwr0iMygVroEpepjMqNcqZIKpk7JWUobSmmlQczIzbIYscPkwtYPH0WP42doijbEognTusMiTX1vRx-GdE5OWtslvDjUOXl_fHi7fy5Xr08v98tV6QSFsdRWtBVTstaKV1Iib5nVUnOXD7EU0CEwKRQTrLIg17wRFmusOV0Ds0wCn5Prve82hq8J02h6nxx2nR0wTMmwuspqLjIIe9DFkFLE1myj7238MUDNLhWzMTkVs0tlN8qpZM3VwdwmZ7s22sH59CdUWoOu6szd7TnMn357jCY5j4PDxkd0o2mC_2fLL65ddNc</recordid><startdate>20061115</startdate><enddate>20061115</enddate><creator>Chen, William Y.C.</creator><creator>Hou, Qing-Hu</creator><creator>Mu, Yan-Ping</creator><general>Elsevier B.V</general><general>Elsevier</general><scope>6I.</scope><scope>AAFTH</scope><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>H8D</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>20061115</creationdate><title>A telescoping method for double summations</title><author>Chen, William Y.C. ; Hou, Qing-Hu ; Mu, Yan-Ping</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c401t-9a4f62758973655e3f2a9593c778a01ece125472426a15b3d4ae8e830b12a2513</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2006</creationdate><topic>Algorithmics. Computability. Computer arithmetics</topic><topic>Applied sciences</topic><topic>Computer science; control theory; systems</topic><topic>Double summation</topic><topic>Exact sciences and technology</topic><topic>Hypergeometric term</topic><topic>Mathematical analysis</topic><topic>Mathematics</topic><topic>Sciences and techniques of general use</topic><topic>Special functions</topic><topic>Theoretical computing</topic><topic>Zeilberger's algorithm</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Chen, William Y.C.</creatorcontrib><creatorcontrib>Hou, Qing-Hu</creatorcontrib><creatorcontrib>Mu, Yan-Ping</creatorcontrib><collection>ScienceDirect Open Access Titles</collection><collection>Elsevier:ScienceDirect:Open Access</collection><collection>Pascal-Francis</collection><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>Aerospace Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Journal of computational and applied mathematics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Chen, William Y.C.</au><au>Hou, Qing-Hu</au><au>Mu, Yan-Ping</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>A telescoping method for double summations</atitle><jtitle>Journal of computational and applied mathematics</jtitle><date>2006-11-15</date><risdate>2006</risdate><volume>196</volume><issue>2</issue><spage>553</spage><epage>566</epage><pages>553-566</pages><issn>0377-0427</issn><eissn>1879-1778</eissn><coden>JCAMDI</coden><abstract>We present a method to prove hypergeometric double summation identities. Given a hypergeometric term
F
(
n
,
i
,
j
)
, we aim to find a difference operator
L
=
a
0
(
n
)
N
0
+
a
1
(
n
)
N
1
+
⋯
+
a
r
(
n
)
N
r
and rational functions
R
1
(
n
,
i
,
j
)
,
R
2
(
n
,
i
,
j
)
such that
LF
=
Δ
i
(
R
1
F
)
+
Δ
j
(
R
2
F
)
. Based on simple divisibility considerations, we show that the denominators of
R
1
and
R
2
must possess certain factors which can be computed from
F
(
n
,
i
,
j
)
. Using these factors as estimates, we may find the numerators of
R
1
and
R
2
by guessing the upper bounds of the degrees and solving systems of linear equations. Our method is valid for the Andrews–Paule identity, Carlitz's identities, the Apéry–Schmidt–Strehl identity, the Graham–Knuth–Patashnik identity, and the Petkovšek–Wilf–Zeilberger identity.</abstract><cop>Amsterdam</cop><pub>Elsevier B.V</pub><doi>10.1016/j.cam.2005.10.010</doi><tpages>14</tpages><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0377-0427 |
| ispartof | Journal of computational and applied mathematics, 2006-11, Vol.196 (2), p.553-566 |
| issn | 0377-0427 1879-1778 |
| language | eng |
| recordid | cdi_proquest_miscellaneous_28672434 |
| source | Elsevier |
| subjects | Algorithmics. Computability. Computer arithmetics Applied sciences Computer science control theory systems Double summation Exact sciences and technology Hypergeometric term Mathematical analysis Mathematics Sciences and techniques of general use Special functions Theoretical computing Zeilberger's algorithm |
| title | A telescoping method for double summations |
| url | http://sfxeu10.hosted.exlibrisgroup.com/loughborough?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-02-24T02%3A20%3A45IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest_cross&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=A%20telescoping%20method%20for%20double%20summations&rft.jtitle=Journal%20of%20computational%20and%20applied%20mathematics&rft.au=Chen,%20William%20Y.C.&rft.date=2006-11-15&rft.volume=196&rft.issue=2&rft.spage=553&rft.epage=566&rft.pages=553-566&rft.issn=0377-0427&rft.eissn=1879-1778&rft.coden=JCAMDI&rft_id=info:doi/10.1016/j.cam.2005.10.010&rft_dat=%3Cproquest_cross%3E28672434%3C/proquest_cross%3E%3Cgrp_id%3Ecdi_FETCH-LOGICAL-c401t-9a4f62758973655e3f2a9593c778a01ece125472426a15b3d4ae8e830b12a2513%3C/grp_id%3E%3Coa%3E%3C/oa%3E%3Curl%3E%3C/url%3E&rft_id=info:oai/&rft_pqid=28672434&rft_id=info:pmid/&rfr_iscdi=true |