Hypertranscendental function

<h2 id="history">History</h2>
<p>The term 'transcendentally transcendental' was introduced by <a href="/facts/E._H._Moore/1lJXH5jn">E. H. Moore</a> in 1896; the term 'hypertranscendental' was introduced by <a href="/facts/D._D._Morduhai-Boltovskoi/50hoX929">D. D. Morduhai-Boltovskoi</a> in 1914.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p>
<h2 id="definition">Definition</h2>
<p>One standard definition (there are slight variants) defines solutions of <a href="/facts/Differential_equation/9MGbE3Ca">differential equations</a> of the form
</p>

F
        
          (
          
            x
            ,
            y
            ,
            
              y
              ′
            
            ,
            ⋯
            ,
            
              y
              
                (
                n
                )
              
            
          
          )
        
        =
        0
      
    
    {\displaystyle F\left(x,y,y',\cdots ,y^{(n)}\right)=0}
  
,
<p>where 
  
    
      
        F
      
    
    {\displaystyle F}
  
 is a polynomial with constant coefficients, as <i>algebraically transcendental</i> or <i>differentially algebraic</i>. Transcendental functions which are not <i>algebraically transcendental</i> are <i>transcendentally transcendental</i>. <a href="/facts/H%25C3%25B6lder%2527s_theorem/zcldpdQ4">Hölder's theorem</a> shows that the <a href="/facts/Gamma_function/SjGQcnyw">gamma function</a> is in this category.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p><p>Hypertranscendental functions usually arise as the solutions to <a href="/facts/Functional_equation/4ovU4CAt">functional equations</a>, for example the <a href="/facts/Gamma_function/SjGQcnyw">gamma function</a>.
</p>
<h2 id="examples">Examples</h2>
<h3>Hypertranscendental functions</h3>
<ul><li>The zeta functions of <a href="/facts/Algebraic_number_field/e0K0TXSh">algebraic number fields</a>, in particular, the <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a></li>
<li>The <a href="/facts/Gamma_function/SjGQcnyw">gamma function</a> (<i>cf.</i> <a href="/facts/H%25C3%25B6lder%2527s_theorem/zcldpdQ4">Hölder's theorem</a>)</li></ul>
<h3>Transcendental but not hypertranscendental functions</h3>
<ul><li>The <a href="/facts/Exponential_function/ewlYxvR2">exponential function</a>, <a href="/facts/Logarithm/QeXaV97q">logarithm</a>, and the <a href="/facts/Trigonometric_function/czej7ur0">trigonometric</a> and <a href="/facts/Hyperbolic_function/Y4Cb5S35">hyperbolic</a> functions.</li>
<li>The <a href="/facts/Generalized_hypergeometric_function/fqXXDHdR">generalized hypergeometric functions</a>, including special cases such as <a href="/facts/Bessel_function/hSyvFPqC">Bessel functions</a> (except some special cases which are algebraic).</li></ul>
<h3>Non-transcendental (algebraic) functions</h3>
<ul><li>All <a href="/facts/Algebraic_function/yT7EKmwF">algebraic functions</a>, in particular <a href="/facts/Polynomial/Lzak8VVx">polynomials</a>.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Hypertranscendental_number/jk9hR7hN">Hypertranscendental number</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li>Loxton, J.H., Poorten, A.J. van der, "<a href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN356261603_0016">A class of hypertranscendental functions</a>", <a href="/facts/Aequationes_Mathematicae/HR3a9EPP">Aequationes Mathematicae</a>, Periodical volume 16</li>
<li><a href="/facts/Kurt_Mahler/8VfW4MWQ">Mahler, K.</a>, "Arithmetische Eigenschaften einer Klasse transzendental-transzendenter Funktionen", Math. Z. 32 (1930) 545-585.</li>
<li>Morduhaĭ-Boltovskoĭ, D. (1949), "On hypertranscendental functions and hypertranscendental numbers", <i>Doklady Akademii Nauk SSSR</i>, New Series (in Russian), 64: 21–24, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0028347">0028347</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>D. D. Mordykhai-Boltovskoi, "On hypertranscendence of the function ξ(x, s)", Izv. Politekh. Inst. Warsaw 2:1-16 (1914), cited in Anatoly A. Karatsuba, S. M. Voronin, The Riemann Zeta-Function, 1992, ISBN 3-11-013170-6, p. 390 <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Morduhaĭ-Boltovskoĭ (1949) - Morduhaĭ-Boltovskoĭ, D. (1949), "On hypertranscendental functions and hypertranscendental numbers", Doklady Akademii Nauk SSSR, New Series (in Russian), 64: 21–24, MR 0028347 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0028347" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0028347</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Eliakim H. Moore, "Concerning Transcendentally Transcendental Functions", Mathematische Annalen 48:1-2:49-74 (1896) doi:10.1007/BF01446334 <a href="/wiki/Eliakim_H._Moore" target="_blank">/wiki/Eliakim_H._Moore</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>R. D. Carmichael, "On Transcendentally Transcendental Functions", Transactions of the American Mathematical Society 14:3:311-319 (July 1913) full text JSTOR 1988599 doi:10.1090/S0002-9947-1913-1500949-2 <a href="/wiki/Robert_Daniel_Carmichael" target="_blank">/wiki/Robert_Daniel_Carmichael</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Lee A. Rubel, "A Survey of Transcendentally Transcendental Functions", The American Mathematical Monthly 96:777-788 (November 1989) JSTOR 2324840 <a href="/wiki/JSTOR_(identifier)" target="_blank">/wiki/JSTOR_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
</ol>

Hypertranscendental function open-in-new

Hypertranscendental function