Asymptotic normality of the Stirling-Whitney-Riordan triangle
Recently, Zhu [34] introduced a Stirling-Whitney-Riordan triangle [Tn,k]n,k?0 satisfying the recurrence Tn,k = (b1k + b2)Tn?1,k?1 + [(2?b1 + a1)k + a2 + ?(b1 + b2)]Tn?1,k + ?(a1 + ?b1)(k + 1)Tn?1,k+1, where initial conditions Tn,k = 0 unless 0 ? k ? n and T0,0 = 1. Denote by Tn = Pnk =0 Tn,k. In thi...
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| Veröffentlicht in: | Filomat 2023, Vol.37 (9), p.2923-2934 |
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| creator | Guo, Wan-Ming Liu, Lily |
| description | Recently, Zhu [34] introduced a Stirling-Whitney-Riordan triangle [Tn,k]n,k?0
satisfying the recurrence Tn,k = (b1k + b2)Tn?1,k?1 + [(2?b1 + a1)k + a2 +
?(b1 + b2)]Tn?1,k + ?(a1 + ?b1)(k + 1)Tn?1,k+1, where initial conditions
Tn,k = 0 unless 0 ? k ? n and T0,0 = 1. Denote by Tn = Pnk =0 Tn,k. In this
paper, we show the asymptotic normality of Tn,k and give an asymptotic
formula of Tn. As applications, we show the asymptotic normality of many
famous combinatorial numbers, such as the Stirling numbers of the second
kind, the Whitney numbers of the second kind, the r-Stirling numbers and the
r-Whitney numbers of the second kind. |
| doi_str_mv | 10.2298/FIL2309923G |
| format | Article |
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satisfying the recurrence Tn,k = (b1k + b2)Tn?1,k?1 + [(2?b1 + a1)k + a2 +
?(b1 + b2)]Tn?1,k + ?(a1 + ?b1)(k + 1)Tn?1,k+1, where initial conditions
Tn,k = 0 unless 0 ? k ? n and T0,0 = 1. Denote by Tn = Pnk =0 Tn,k. In this
paper, we show the asymptotic normality of Tn,k and give an asymptotic
formula of Tn. As applications, we show the asymptotic normality of many
famous combinatorial numbers, such as the Stirling numbers of the second
kind, the Whitney numbers of the second kind, the r-Stirling numbers and the
r-Whitney numbers of the second kind.</description><identifier>ISSN: 0354-5180</identifier><identifier>EISSN: 2406-0933</identifier><identifier>DOI: 10.2298/FIL2309923G</identifier><language>eng</language><ispartof>Filomat, 2023, Vol.37 (9), p.2923-2934</ispartof><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c228t-4d2ead4f7a254bb2999c41ce8311797cea2ecb41543321c03e89c7dd5e30cd463</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,776,780,4010,27900,27901,27902</link.rule.ids></links><search><creatorcontrib>Guo, Wan-Ming</creatorcontrib><creatorcontrib>Liu, Lily</creatorcontrib><title>Asymptotic normality of the Stirling-Whitney-Riordan triangle</title><title>Filomat</title><description>Recently, Zhu [34] introduced a Stirling-Whitney-Riordan triangle [Tn,k]n,k?0
satisfying the recurrence Tn,k = (b1k + b2)Tn?1,k?1 + [(2?b1 + a1)k + a2 +
?(b1 + b2)]Tn?1,k + ?(a1 + ?b1)(k + 1)Tn?1,k+1, where initial conditions
Tn,k = 0 unless 0 ? k ? n and T0,0 = 1. Denote by Tn = Pnk =0 Tn,k. In this
paper, we show the asymptotic normality of Tn,k and give an asymptotic
formula of Tn. As applications, we show the asymptotic normality of many
famous combinatorial numbers, such as the Stirling numbers of the second
kind, the Whitney numbers of the second kind, the r-Stirling numbers and the
r-Whitney numbers of the second kind.</description><issn>0354-5180</issn><issn>2406-0933</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2023</creationdate><recordtype>article</recordtype><recordid>eNpNz7tKBDEUgOEgCo6rlS8wvURPzslcUlgsi7suDAhesBwySWY3MpclSTNvL6KF1d_98DF2K-AeUdUP232DBEoh7c5YhhJKDoronGVAheSFqOGSXcX4BSCxlFXGHtdxGU9pTt7k0xxGPfi05HOfp6PL35IPg58O_PPo0-QW_urnYPWUp-D1dBjcNbvo9RDdzV9X7GP79L555s3Lbr9ZN9wg1olLi05b2VcaC9l1qJQyUhhXkxCVqozT6EwnRSGJUBggVytTWVs4AmNlSSt29_s1YY4xuL49BT_qsLQC2h95-09O39WyS-Q</recordid><startdate>2023</startdate><enddate>2023</enddate><creator>Guo, Wan-Ming</creator><creator>Liu, Lily</creator><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>2023</creationdate><title>Asymptotic normality of the Stirling-Whitney-Riordan triangle</title><author>Guo, Wan-Ming ; Liu, Lily</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c228t-4d2ead4f7a254bb2999c41ce8311797cea2ecb41543321c03e89c7dd5e30cd463</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2023</creationdate><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Guo, Wan-Ming</creatorcontrib><creatorcontrib>Liu, Lily</creatorcontrib><collection>CrossRef</collection><jtitle>Filomat</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Guo, Wan-Ming</au><au>Liu, Lily</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Asymptotic normality of the Stirling-Whitney-Riordan triangle</atitle><jtitle>Filomat</jtitle><date>2023</date><risdate>2023</risdate><volume>37</volume><issue>9</issue><spage>2923</spage><epage>2934</epage><pages>2923-2934</pages><issn>0354-5180</issn><eissn>2406-0933</eissn><abstract>Recently, Zhu [34] introduced a Stirling-Whitney-Riordan triangle [Tn,k]n,k?0
satisfying the recurrence Tn,k = (b1k + b2)Tn?1,k?1 + [(2?b1 + a1)k + a2 +
?(b1 + b2)]Tn?1,k + ?(a1 + ?b1)(k + 1)Tn?1,k+1, where initial conditions
Tn,k = 0 unless 0 ? k ? n and T0,0 = 1. Denote by Tn = Pnk =0 Tn,k. In this
paper, we show the asymptotic normality of Tn,k and give an asymptotic
formula of Tn. As applications, we show the asymptotic normality of many
famous combinatorial numbers, such as the Stirling numbers of the second
kind, the Whitney numbers of the second kind, the r-Stirling numbers and the
r-Whitney numbers of the second kind.</abstract><doi>10.2298/FIL2309923G</doi><tpages>12</tpages><oa>free_for_read</oa></addata></record> |
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| source | Jstor Complete Legacy; Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals |
| title | Asymptotic normality of the Stirling-Whitney-Riordan triangle |
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