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Keywords = Blissard umbral approach

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16 pages, 1881 KiB  
Article
A Family of the r-Associated Stirling Numbers of the Second Kind and Generalized Bernoulli Polynomials
by Paolo Emilio Ricci, Rekha Srivastava and Pierpaolo Natalini
Axioms 2021, 10(3), 219; https://doi.org/10.3390/axioms10030219 - 9 Sep 2021
Cited by 5 | Viewed by 3242
Abstract
In this article, we derive representation formulas for a class of r-associated Stirling numbers of the second kind and examine their connections with a class of generalized Bernoulli polynomials. Herein, we use the Blissard umbral approach and the familiar Bell polynomials. Links [...] Read more.
In this article, we derive representation formulas for a class of r-associated Stirling numbers of the second kind and examine their connections with a class of generalized Bernoulli polynomials. Herein, we use the Blissard umbral approach and the familiar Bell polynomials. Links with available literature on this subject are also pointed out. The extension to the bivariate case is discussed. Full article
(This article belongs to the Collection Mathematical Analysis and Applications)
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Figure 1
<p><span class="html-italic">Numbers</span><math display="inline"> <semantics> <mrow> <mspace width="0.277778em"/> <msubsup> <mi>B</mi> <mi>n</mi> <mrow> <mo>[</mo> <mn>1</mn> <mo>,</mo> <mi>k</mi> <mo>]</mo> </mrow> </msubsup> <mo>;</mo> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>;</mo> <mspace width="0.166667em"/> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>10</mn> <mo>.</mo> </mrow> </semantics> </math></p>
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<p><span class="html-italic">Numbers</span><math display="inline"> <semantics> <mrow> <mspace width="0.277778em"/> <msubsup> <mi>B</mi> <mi>n</mi> <mrow> <mo>[</mo> <mn>2</mn> <mo>,</mo> <mi>k</mi> <mo>]</mo> </mrow> </msubsup> <mo>;</mo> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>;</mo> <mspace width="0.166667em"/> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>10</mn> <mo>.</mo> </mrow> </semantics> </math></p>
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<p><span class="html-italic">Numbers</span><math display="inline"> <semantics> <mrow> <mspace width="0.277778em"/> <msubsup> <mi>B</mi> <mi>n</mi> <mrow> <mo>[</mo> <mn>3</mn> <mo>,</mo> <mi>k</mi> <mo>]</mo> </mrow> </msubsup> <mo>;</mo> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>;</mo> <mspace width="0.166667em"/> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>10</mn> <mo>.</mo> </mrow> </semantics> </math></p>
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<p><span class="html-italic">Numbers</span><math display="inline"> <semantics> <mrow> <mspace width="0.277778em"/> <msubsup> <mi>B</mi> <mi>n</mi> <mrow> <mo>[</mo> <mn>4</mn> <mo>,</mo> <mi>k</mi> <mo>]</mo> </mrow> </msubsup> <mo>;</mo> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>;</mo> <mspace width="0.166667em"/> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>10</mn> <mo>.</mo> </mrow> </semantics> </math></p>
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<p><span class="html-italic">Numbers</span><math display="inline"> <semantics> <mrow> <mi>S</mi> <mo>(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mi>k</mi> <mo>,</mo> <mi>k</mi> <mo>;</mo> <mn>2</mn> <mo>)</mo> <mo>,</mo> <mspace width="0.166667em"/> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>;</mo> <mspace width="0.166667em"/> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>10</mn> <mo>.</mo> </mrow> </semantics> </math></p>
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<p><span class="html-italic">Numbers</span><math display="inline"> <semantics> <mrow> <mi>S</mi> <mo>(</mo> <mi>n</mi> <mo>+</mo> <mn>3</mn> <mi>k</mi> <mo>,</mo> <mi>k</mi> <mo>;</mo> <mn>3</mn> <mo>)</mo> <mo>,</mo> <mspace width="0.166667em"/> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>;</mo> <mspace width="0.166667em"/> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>10</mn> <mo>.</mo> </mrow> </semantics> </math></p>
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<p><span class="html-italic">Numbers</span><math display="inline"> <semantics> <mrow> <mi>S</mi> <mo>(</mo> <mi>n</mi> <mo>+</mo> <mn>4</mn> <mi>k</mi> <mo>,</mo> <mi>k</mi> <mo>;</mo> <mn>4</mn> <mo>)</mo> <mo>,</mo> <mspace width="0.166667em"/> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>;</mo> <mspace width="0.166667em"/> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>10</mn> <mo>.</mo> </mrow> </semantics> </math></p>
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<p><span class="html-italic">Numbers</span><math display="inline"> <semantics> <mrow> <mi>S</mi> <mo>(</mo> <mi>n</mi> <mo>+</mo> <mn>5</mn> <mi>k</mi> <mo>,</mo> <mi>k</mi> <mo>;</mo> <mn>5</mn> <mo>)</mo> <mo>,</mo> <mspace width="0.166667em"/> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>;</mo> <mspace width="0.166667em"/> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mn>10</mn> <mo>.</mo> </mrow> </semantics> </math></p>
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