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- W3121405734 abstract "Abstract In this paper, by means of the summation property to the Riordan array, we derive some identities involving generalized harmonic, hyperharmonic and special numbers. For example, for n ≥ 0, <m:math xmlns:m=http://www.w3.org/1998/Math/MathML display=block> <m:mrow> <m:munderover> <m:mo>∑</m:mo> <m:mrow> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mi>n</m:mi> </m:munderover> <m:mrow> <m:mfrac> <m:mrow> <m:msub> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mi>k</m:mi> </m:msub> </m:mrow> <m:mrow> <m:mi>k</m:mi> <m:mo>!</m:mo> </m:mrow> </m:mfrac> <m:mi>H</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>.</m:mo> <m:mi>k</m:mi> <m:mo>,</m:mo> <m:mi>α</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>α</m:mi> <m:mi>H</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>α</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>-</m:mo> <m:mi>H</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>α</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> sumlimits_{k = 0}^n {{{{B_k}} over {k!}}Hleft( {n.k,alpha } right) = alpha Hleft( {n + 1,1,alpha } right) - Hleft( {n,1,alpha } right)} , and for n > r ≥ 0, <m:math xmlns:m=http://www.w3.org/1998/Math/MathML display=block> <m:mrow> <m:munderover> <m:mo>∑</m:mo> <m:mrow> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:mi>r</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:munderover> <m:mrow> <m:msup> <m:mrow> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mi>k</m:mi> </m:msup> <m:mfrac> <m:mrow> <m:mi>s</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>k</m:mi> <m:mo>,</m:mo> <m:mi>r</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi>r</m:mi> <m:mo>!</m:mo> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mi>α</m:mi> </m:mrow> <m:mi>k</m:mi> </m:msup> <m:mi>k</m:mi> <m:mo>!</m:mo> </m:mrow> </m:mfrac> <m:msub> <m:mrow> <m:mi>H</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mi>k</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mi>α</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mi>r</m:mi> </m:msup> <m:mi>H</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mi>r</m:mi> <m:mo>,</m:mo> <m:mi>α</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> sumlimits_{k = r}^{n - 1} {{{left( { - 1} right)}^k}{{sleft( {k,r} right)r!} over {{alpha ^k}k!}}{H_{n - k}}left( alpha right) = {{left( { - 1} right)}^r}Hleft( {n,r,alpha } right)} , where Bernoulli numbers Bn and Stirling numbers of the first kind s ( n , r )." @default.
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- W3121405734 date "2021-01-01" @default.
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- W3121405734 title "On identities involving generalized harmonic, hyperharmonic and special numbers with Riordan arrays" @default.
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- W3121405734 doi "https://doi.org/10.1515/spma-2020-0111" @default.
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