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22                        <a href="#sect-publ" class="w3-bar-item w3-button w3-round-large w3-hover-teal">Publications</a>
23                        <a href="#sect-form" class="w3-bar-item w3-button w3-round-large w3-hover-teal">Formulas</a>
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30        <section id="sect-home">
31            <h1>Homepage</h1>
32            Bernd C. Kellner<br>
33            Göttingen, Germany<br>
34            Email: bk (at) bernoulli.org<br><br>
35        </section>
36
37        <section>
38            <h1>Thesis</h1>
39            <cite>Über irreguläre Paare höherer Ordnungen</cite>. <br>
40            Diplomarbeit. Mathematisches Institut der Georg-August-Universität zu Göttingen, Germany, 2002.<br>
41            Online: <a class="tooltip-top" href="irrpairord.pdf">irrpairord.pdf<span class="tooltip-text">Download</span></a>&nbsp; (891 KB)<br><br>
42        </section>
43
44        <section id="sect-publ">
45            <h1>Publications</h1>
46            <ol>
47                <li>
48                    <cite>On irregular prime power divisors of the Bernoulli numbers</cite>,
49                    Math. Comp. <b>76</b> (2007), <span class="cite-page">405&ndash;441</span>.<br>
50                    <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1183.11012" rel="noopener">1183.11012</a></span> &ensp;
51                    <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.1090/S0025-5718-06-01887-4" rel="noopener">10.1090/S0025-5718-06-01887-4</a></span>
52                </li>
53                <li>
54                    <cite>On asymptotic constants related to products of Bernoulli numbers and factorials</cite>,
55                    Integers <b>9</b> (2009), Article A08, <span class="cite-page">83&ndash;106</span>.<br>
56                    <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1163.11014" rel="noopener">1163.11014</a></span> &ensp;
57                    <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.1515/INTEG.2009.009" rel="noopener">10.1515/INTEG.2009.009</a></span> &ensp;
58                    <span class="cite-link"><b>Link</b>: <a class="w3-hover-shadow" target="_blank" href="http://www.integers-ejcnt.org/vol9.html" rel="noopener">Integers Vol. 9</a></span>
59                </li>
60                <li>
61                    <cite>On stronger conjectures that imply the Erdős&ndash;Moser conjecture</cite>,
62                    J. Number Theory <b>131</b> (2011), <span class="cite-page">1054&ndash;1061</span>.<br>
63                    <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1267.11031" rel="noopener">1267.11031</a></span> &ensp;
64                    <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.1016/j.jnt.2011.01.004" rel="noopener">10.1016/j.jnt.2011.01.004</a></span>
65                </li>
66                <li>
67                    <cite>On quotients of Riemann zeta values at odd and even integer arguments</cite>,
68                    J. Number Theory <b>133</b> (2013), <span class="cite-page">2684&ndash;2698</span>.<br>
69                    <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1290.11118" rel="noopener">1290.11118</a></span> &ensp;
70                    <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.1016/j.jnt.2013.02.008" rel="noopener">10.1016/j.jnt.2013.02.008</a></span>
71                </li>
72                <li>
73                    <cite>Identities between polynomials related to Stirling and harmonic numbers</cite>,
74                    Integers <b>14</b> (2014), Article A54, <span class="cite-page">1&ndash;22</span>.<br>
75                    <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1315.11018" rel="noopener">1315.11018</a></span> &ensp;
76                    <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.5281/zenodo.10452538" rel="noopener">10.5281/zenodo.10452538</a></span> &ensp;
77                    <span class="cite-link"><b>Link</b>: <a class="w3-hover-shadow" target="_blank" href="http://www.integers-ejcnt.org/vol14.html" rel="noopener">Integers Vol. 14</a></span>
78                </li>
79                <li>
80                    <cite>The topology of Stein fillable manifolds in high dimensions II (with an appendix by <b>Bernd C. Kellner</b>)</cite>,
81                    Geom. Topol. <b>19</b> (2015), <span class="cite-page">2995&ndash;3030</span>.<br>
82                    Authors: <b>Jonathan Bowden</b>, <b>Diarmuid Crowley</b>, <b>András I. Stipsicz</b>. Appendix by <b>Bernd C. Kellner</b>.<br>
83                    <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1380.32016" rel="noopener">1380.32016</a></span> &ensp;
84                    <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.2140/gt.2015.19.2995" rel="noopener">10.2140/gt.2015.19.2995</a></span>
85                </li>
86                <li>
87                    <cite>On a product of certain primes</cite>,
88                    J. Number Theory <b>179</b> (2017), <span class="cite-page">126&ndash;141</span>.<br>
89                    <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1418.11045" rel="noopener">1418.11045</a></span> &ensp;
90                    <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.1016/j.jnt.2017.03.020" rel="noopener">10.1016/j.jnt.2017.03.020</a></span>
91                </li>
92                <li>
93                    <cite>Power-sum denominators</cite>,
94                    Amer. Math. Monthly <b>124</b> (2017), <span class="cite-page">695&ndash;709</span>.<br>
95                    Co-author: <b>Jonathan Sondow</b><br>
96                    <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1391.11052" rel="noopener">1391.11052</a></span> &ensp;
97                    <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.4169/amer.math.monthly.124.8.695" rel="noopener">10.4169/amer.math.monthly.124.8.695</a></span>
98                </li>
99                <li>
100                    <cite>The denominators of power sums of arithmetic progressions</cite>,
101                    Integers <b>18</b> (2018), Article A95, <span class="cite-page">1&ndash;17</span>.<br>
102                    Co-author: <b>Jonathan Sondow</b><br>
103                    <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1423.11029" rel="noopener">1423.11029</a></span> &ensp;
104                    <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.5281/zenodo.10682734" rel="noopener">10.5281/zenodo.10682734</a></span> &ensp;
105                    <span class="cite-link"><b>Link</b>: <a class="w3-hover-shadow" target="_blank" href="http://www.integers-ejcnt.org/vol18.html" rel="noopener">Integers Vol. 18</a></span>
106                </li>
107                <li>
108                    <cite>On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits</cite>,
109                    Integers <b>21</b> (2021), Article A52, <span class="cite-page">1&ndash;21</span>.<br>
110                    Co-author: <b>Jonathan Sondow</b><br>
111                    <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1479.11028" rel="noopener">1479.11028</a></span> &ensp;
112                    <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.5281/zenodo.10816833" rel="noopener">10.5281/zenodo.10816833</a></span> &ensp;
113                    <span class="cite-link"><b>Link</b>: <a class="w3-hover-shadow" target="_blank" href="http://www.integers-ejcnt.org/vol21.html" rel="noopener">Integers Vol. 21</a></span>
114                </li>
115                <li>
116                    <cite>On (self-) reciprocal Appell polynomials: symmetry and Faulhaber-type polynomials</cite>,
117                    Integers <b>21</b> (2021), Article A119, <span class="cite-page">1&ndash;19</span>.<br>
118                    <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1490.11033" rel="noopener">1490.11033</a></span> &ensp;
119                    <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.5281/zenodo.10842328" rel="noopener">10.5281/zenodo.10842328</a></span> &ensp;
120                    <span class="cite-link"><b>Link</b>: <a class="w3-hover-shadow" target="_blank" href="http://www.integers-ejcnt.org/vol21.html" rel="noopener">Integers Vol. 21</a></span>
121                </li>
122                <li>
123                    <cite>Shifted sums of the Bernoulli numbers, reciprocity, and denominators</cite>,
124                    Rend. Mat. Appl. VII. Ser. <b>43</b> (2022), <span class="cite-page">151&ndash;163</span>.<br>
125                    <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1503.11047" rel="noopener">1503.11047</a></span> &ensp;
126                    <span class="cite-link"><b>Link</b>: <a class="w3-hover-shadow" target="_blank" href="https://www1.mat.uniroma1.it/ricerca/rendiconti/43_2_(2022)_151-163.html" rel="noopener">Rend. Mat. Appl. VII. Ser. Vol. 43</a></span>
127                </li>
128                <li>
129                    <cite>On primary Carmichael numbers</cite>,
130                    Integers <b>22</b> (2022), Article A38, <span class="cite-page">1&ndash;39</span>.<br>
131                    <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1495.11008" rel="noopener">1495.11008</a></span> &ensp;
132                    <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.5281/zenodo.10963985" rel="noopener">10.5281/zenodo.10963985</a></span> &ensp;
133                    <span class="cite-link"><b>Link</b>: <a class="w3-hover-shadow" target="_blank" href="http://www.integers-ejcnt.org/vol22.html" rel="noopener">Integers Vol. 22</a></span>
134                </li>
135                <li>
136                    <cite>Faulhaber polynomials and reciprocal Bernoulli polynomials</cite>,
137                    Rocky Mountain J. Math. <b>53</b> (2023), <span class="cite-page">119&ndash;151</span>.<br>
138                    <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1529.11039" rel="noopener">1529.11039</a></span> &ensp;
139                    <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.1216/rmj.2023.53.119" rel="noopener">10.1216/rmj.2023.53.119</a></span>
140                </li>
141               <li>
142                    <cite>On the nonintegrality of certain generalized binomial sums</cite>,
143                    Integers <b>23</b> (2023), Article A45, <span class="cite-page">1&ndash;24</span>.<br>
144                    <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1548.11050" rel="noopener">1548.11050</a></span> &ensp;
145                    <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.5281/zenodo.8099775" rel="noopener">10.5281/zenodo.8099775</a></span> &ensp;
146                    <span class="cite-link"><b>Link</b>: <a class="w3-hover-shadow" target="_blank" href="http://www.integers-ejcnt.org/vol23.html" rel="noopener">Integers Vol. 23</a></span>
147                </li>
148               <li>
149                    <cite>On the finiteness of Bernoulli polynomials whose derivative has only integral coefficients</cite>,
150                    J. Integer Seq. <b>27</b> (2024), Article 24.2.8, <span class="cite-page">1&ndash;11</span>.<br>
151                    <span class="cite-link"><b>Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://zbmath.org/1541.11023" rel="noopener">1541.11023</a></span> &ensp;
152                    <span class="cite-link"><b>Link</b>: <a class="w3-hover-shadow" target="_blank" href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Kellner/kell2.html" rel="noopener">Integer Seq. Vol. 27</a></span>
153                </li>
154               <li>
155                    <cite>Asymptotic products of binomial and multinomial coefficients revisited</cite>,
156                    Integers <b>24</b> (2024), Article A59, <span class="cite-page">1&ndash;10</span>.<br>
157                    <span class="cite-link"><b>DOI</b>: <a class="w3-hover-shadow" target="_blank" href="https://doi.org/10.5281/zenodo.12167556" rel="noopener">10.5281/zenodo.12167556</a></span> &ensp;
158                    <span class="cite-link"><b>Link</b>: <a class="w3-hover-shadow" target="_blank" href="http://www.integers-ejcnt.org/vol24.html" rel="noopener">Integers Vol. 24</a></span>
159                </li>
160             </ol>
161        </section>
162
163        <section>
164            <h1>Preprints</h1>
165            <ol type="A">
166                <li>
167                    <cite>The equation <span class="math"><span class="math-oper">denom</span>(<span class="math-var">B<sub class="math-var">n</sub></span>) &equals; <span class="math-var">n</span></span> has only one solution</cite>.
168                    Online: <a class="tooltip-top" href="denombneqn.pdf">denombneqn.pdf<span class="tooltip-text">Download</span></a>
169                </li>
170                <li>
171                    <cite>On the <span class="math math-var">p</span>-divisibility of the sequence 
172                        <span class="math"><span class="math-var">B</span><sub class="math-var">lp<sup>r</sup></sub> / <span class="math-var">lp</span><sup class="math-var">r</sup></span></cite>.
173                    Online: <a class="tooltip-top" href="pdivbn.pdf">pdivbn.pdf<span class="tooltip-text">Download</span></a>
174                </li>
175                <li>
176                    <cite>Distribution modulo one and denominators of the Bernoulli polynomials</cite>,
177                    <span class="cite-link"><b>arXiv</b>: <a class="w3-hover-shadow" target="_blank" href="https://arxiv.org/abs/1708.07119" rel="noopener">1708.07119</a></span>.<br>
178                </li>
179                <li>
180                    My preprints on <a class="w3-hover-shadow" target="_blank" href="https://arxiv.org/find/math/1/au:+Kellner_B/0/1/0/all/0/1" rel="noopener"><b>arXiv.org</b></a>
181                </li>
182            </ol>
183        </section>
184
185        <section id="sect-form">
186            <h1>Formulas</h1>
187            <h2 class="w3-round-large">Asymptotic products</h2>
188            <h3>Product of values of the gamma function</h3>
189            <p>Product of increasing powers of gamma values at rational arguments [<span class="cite-ref">2, Cor.&nbsp;18, p.&nbsp;93</span>]:</p>
190            <p class="w3-center"><img src="img/gamma1.png" width="362" height="81" alt="gamma1"></p>
191            <p>with the constants</p>
192            <p class="w3-center"><img src="img/gamma2.png" width="268" height="129" alt="gamma2"></p>
193            <p>where <span class="math">&gamma;</span> is Euler's constant [<span class="cite-oeis">OEIS <a class="w3-hover-shadow " target="_blank" href="https://www.oeis.org/A001620" rel="noopener">A001620</a></span>] 
194                and <span class="math"><img class="math-base" src="img/cca.png" width="14" height="14" alt="&Ascr;"></span> is the Glaisher-Kinkelin constant [<span class="cite-oeis">OEIS <a class="w3-hover-shadow" target="_blank" href="https://www.oeis.org/A074962" rel="noopener">A074962</a></span>].</p>
195
196            <h3>Product of factorials</h3>
197            <p>Special product ([<span class="cite-ref">2, Thms.&nbsp;12,&nbsp;13, p.&nbsp;88</span>] for <span class="math"><span class="math-var">k</span> &equals; <span class="math-num">5</span></span>):</p>
198            <p class="w3-center"><img src="img/factorial1.png" width="464" height="64" alt="factorial1"></p>
199            <p>with the constant</p>
200            <p class="w3-center"><img src="img/factorial2.png" width="453" height="101" alt="factorial2"></p>
201            <p>where <span class="math">(<span class="math-num">1</span>&plus;&radic;<span class="math-num">5</span>)/<span class="math-num">2</span></span> is the golden ratio [<span class="cite-oeis">OEIS <a class="w3-hover-shadow" target="_blank" href="https://www.oeis.org/A001622" rel="noopener">
201A001622</a></span>].</p> 
202
203            <h3>Product of the Bernoulli numbers</h3>
204            <p>Product of (divided) Bernoulli numbers [<span class="cite-ref">2, Thm.&nbsp;21, p.&nbsp;95</span>]:</p>
205            <p class="w3-center"><img src="img/bnprod1.png" width="417" height="145" alt="bnprod1"></p>
206            <p>with the constants</p>
207            <p class="w3-center"><img src="img/bnprod2.png" width="360" height="141" alt="bnprod2"></p>
208            <p>where <span class="math"><img class="math-base" src="img/ccz.png" width="13" height="12" alt="&Zscr;"></span> is the product over all Riemann zeta values at even positive integer arguments [<span class="cite-oeis">OEIS <a class="w3-hover-shadow" target="_blank" href="https://www.oeis.org/A080729" rel="noopener">A080729</a></span>].</p>
209
210            <h2 class="w3-round-large">Number theory</h2>
211            <h3>Divisor function</h3>
212            <p>Recurrence formula for the divisor function for even <span class="math"><span class="math-var">n</span> &geq; <span class="math-num">8</span></span> [<span class="cite-ref">Thesis, Satz&nbsp;1.1.5, p.&nbsp;8</span>]:</p>
213            <p class="w3-center"><img src="img/divisor1.png" width="756" height="86" alt="divisor1"></p>
214            <p>with the convolution function</p>
215            <p class="w3-center"><img src="img/divisor2.png" width="281" height="67" alt="divisor2"></p>
216            <p>The first case <span class="math"><span class="math-var">n</span> &equals; <span class="math-num">8</span></span> is known as Hurwitz's identity (1881)</p>
217            <p class="w3-center"><img src="img/divisor3.png" width="192" height="24" alt="divisor3"></p>
218
219            <h3>Identities with Stirling and harmonic numbers</h3>
220            <p>Define the polynomials</p>
221            <p class="w3-center"><img src="img/identsh1.png" width="568" height="65" alt="identsh1"></p>
222            <p>composed of Stirling numbers of the second kind <span class="math"><span class="math-bold">S</span><sub class="math-num">2</sub>(<span class="math-var">n</span>,<span class="math-var">k</span>)</span> and harmonic numbers <span class="math"><span class="math-bold">H</span><sub class="math-var">k</sub></span> by</p>
223            <p class="w3-center"><img src="img/identsh2.png" width="333" height="68" alt="identsh2"></p>
224            <p>The Genocchi numbers <span class="math"><span class="math-var">G</span><sub class="math-var">n</sub></span> [<span class="cite-oeis">OEIS <a class="w3-hover-shadow" target="_blank" href="https://www.oeis.org/A036968" rel="noopener">A036968</a></span>] are related to the Bernoulli numbers by</p>
225            <p class="w3-center"><img src="img/identsh3.png" width="280" height="24" alt="identsh3"></p>
226            <p>The second identity of <span class="math"><span class="math-var">B</span><sub class="math-var">n</sub></span> is due to Worpitzky (1883):</p>
227            <p class="w3-center"><img src="img/identsh4.png" width="367" height="65" alt="identsh4"></p>
228            <p>The <q>central value</q> is known to be</p>
229            <p class="w3-center"><img src="img/identsh5.png" width="162" height="51" alt="identsh5"></p>
230            <p>Involving the harmonic numbers leads to the identities [<span class="cite-ref">5, Thm.&nbsp;1.3, p.&nbsp;3</span>]:</p>
231            <p class="w3-center"><img src="img/identsh6.png" width="465" height="65" alt="identsh6"></p>
232            <p>and</p>
233            <p class="w3-center"><img src="img/identsh7.png" width="545" height="120" alt="identsh7"></p>
234
235            <h3>Riemann zeta function</h3>
236            <p>Quotient of Riemann zeta values [<span class="cite-ref">4, Thm.&nbsp;1.3, p.&nbsp;2687</span>]: If <span class="math"><span class="math-var">n</span> &geq; <span class="math-num">2</span></span> is even, then</p>
237            <p class="w3-center"><img src="img/zetaquot1.png" width="458" height="58" alt="zetaquot1"></p>
238            <p>with</p>
239            <p class="w3-center"><img src="img/zetaquot2.png" width="226" height="84" alt="zetaquot2"></p>
240            <p>where <span class="math"><img class="math-base" src="img/cclstar.png" width="18" height="13" alt="&Lscr;&Star;"></span> is a linear functional, which is connected with a special Dirichlet series,
241                and <span class="math"><img src="img/lfpn.png" width="16" height="13" alt="&pfr;n"></span> is a certain monic polynomial of degree <span class="math math-var">n</span>.
242                Moreover, if <span class="math"><span class="math-var">n</span> &equals; <span class="math-var">p</span> &plus; <span class="math-num">1</span></span> with <span class="math math-var">p</span> an odd prime, 
243                then <span class="math"><img src="img/lfqn.png" width="16" height="13" alt="&qfr;n"></span> is an Eisenstein polynomial and therefore irreducible over <span class="math">&Zopf;[<span class="math-var">x</span>]</span>.</p>
244
245            <h3>Conjecture on the structure of the Bernoulli numbers</h3>
246            <p>Under the assumption that every <span class="math math-var">p</span>-adic zeta function 
247                <span class="math"><span class="math-var">&zeta;</span><sub>(<span class="math-var">p</span>,<span class="math-var">&ell;</span>)</sub>(<span class="math-var">s</span>)</span> has a unique simple zero 
248                <span class="math"><span class="math-var">&xi;</span><sub>(<span class="math-var">p</span>,<span class="math-var">&ell;</span>)</sub></span>
249                in case <span class="math">(<span class="math-var">p</span>,<span class="math-var">&ell;</span>)</span> is an irregular pair, 
250                one has for even <span class="math"><span class="math-var">n</span> &geq; <span class="math-num">2</span></span> [<span class="cite-ref">1, Thm.&nbsp;4.9, p.&nbsp;16</span>]:</p>
251            <p class="w3-center"><img src="img/bnstruct1.png" width="518" height="88" alt="bnstruct1"></p>
252            <p>where</p>
253            <p class="w3-center"><img src="img/bnstruct2.png" width="415" height="133" alt="bnstruct2"></p>
254            <p>The denominator can be described by poles (always lying at <span class="math math-num">0</span>) and the numerator by zeros of <span class="math math-var">p</span>-adic zeta functions. Equivalently, the formula reads for the Bernoulli numbers:</p>
255            <p class="w3-center"><img src="img/bnstruct3.png" width="653" height="80" alt="bnstruct3"></p>
256            <p>Moreover, the formulas are valid for all irregular pairs <span class="math">(<span class="math-var">p</span>,<span class="math-var">&ell;</span>)</span> with</p>
257            <p class="w3-center"><img src="img/bnstruct4.png" width="234" height="25" alt="bnstruct4"></p>
258
259            <h3>Denominators of the Bernoulli polynomials</h3>
260            <p>The denominator of <span class="math"><span class="math-var">B</span><sub class="math-var">n</sub>(<span class="math-var">x</span>) &minus; <span class="math-var">B</span><sub class="math-var">n</sub></span>, 
261                the <span class="math math-var">n</span>th Bernoulli polynomial without constant term, is given by the remarkable formula ([<span class="cite-ref">7</span>], [<span class="cite-ref">8</span>] with Jonathan Sondow, [<span class="cite-oeis">OEIS <a class="w3-hover-shadow" target="_blank" href="https://www.oeis.org/A195441" rel="noopener">A195441</a></span>])</p>
262            <p class="w3-center"><img src="img/bpdenom1.png" width="311" height="56" alt="bpdenom1"></p>
263            <p>where <span class="math"><span class="math-var">s</span><sub class="math-var">p</sub>(<span class="math-var">n</span>)</span> denotes the sum of the base-<span class="math math-var">p</span> digits of <span class="math math-var">n</span>.
264                The finite product can be written with explicit sharp bounds as</p>
265            <p class="w3-center"><img src="img/bpdenom2.png" width="311" height="83" alt="bpdenom2"></p>
266            <p>where</p>
267            <p class="w3-center"><img src="img/bpdenom3.png" width="219" height="58" alt="bpdenom3"></p>
268            <p>The denominator of the <span class="math math-var">n</span>th Bernoulli polynomial <span class="math"><span class="math-var">B</span><sub class="math-var">n</sub>(<span class="math-var">x</span>)</span> can be described by a similar formula ([<span class="cite-ref">9</span>] with Jonathan Sondow):</p>
269            <p class="w3-center"><img src="img/bpdenom5.png" width="372" height="74" alt="bpdenom5"></p>            
270            <p>If <span class="math">{<span style="vertical-align: middle">&centerdot;</span>}</span> denotes the fractional part and <span class="math"><span class="math-var">n</span> &geq; <span class="math-num">1</span></span> is a fixed integer, 
271                then there is the surprising relation [<span class="cite-ref">C</span>]:</p>
272            <p class="w3-center"><img src="img/bpdenom4.png" width="495" height="65" alt="bpdenom4"></p>
273        </section>
274
275        <section id="sect-link">
276            <h1>Links</h1>
277            <ul>
278                <li><b>B. C. Kellner</b>: <a class="w3-hover-shadow" target="_blank" href="https://www.bernoulli.org" rel="noopener">The Bernoulli number page</a></li>
279                <li><b>K. Matthews</b>: <a class="w3-hover-shadow" target="_blank" href="http://www.numbertheory.org" rel="noopener">Number Theory Web</a></li>
280                <li><b>OEIS</b>: <a class="w3-hover-shadow" target="_blank" href="https://oeis.org" rel="noopener">The On-Line Encyclopedia of Integer Sequences</a></li>
281                <li><b>
281Zbl</b>: <a class="w3-hover-shadow" target="_blank" href="https://www.zbmath.org" rel="noopener">zbMATH Open</a></li>
282            </ul>
283        </section>
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