SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>Prabhakar function</title>

</head>

<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Prabhakar function</span></span>

</h1>

</header>

<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p><b>Prabhakar function</b> is a certain <a href="Special_function" class="mw-redirect" title="Special function">special function</a> in mathematics introduced by the Indian mathematician <a href="Tilak_Raj_Prabhakar" title="Tilak Raj Prabhakar">Tilak Raj Prabhakar</a> in a paper published in 1971.<sup id="cite_ref-Tilak_1-0" class="reference"><a href="#cite_note-Tilak-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The function is a three-parameter generalization of the well known two-parameter <a href="Mittag-Leffler_function" title="Mittag-Leffler function">Mittag-Leffler function</a> in mathematics. The function was originally introduced to solve certain classes of <a href="Integral_equation" title="Integral equation">integral equations</a>. Later the function was found to have applications in the theory of <a href="Fractional_calculus" title="Fractional calculus">fractional calculus</a> and also in certain areas of physics.<sup id="cite_ref-Andrea_2-0" class="reference"><a href="#cite_note-Andrea-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>The one-parameter and two-parameter Mittag-Leffler functions are defined first. Then the definition of the three-parameter Mittag-Leffler function, the Prabhakar function, is presented. In the following definitions, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma (z)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (z)}</annotation>

</semantics>

</math></span><img src="./11ca17f880240539116aac7e6326909299e2a080.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.35ex; height:2.843ex;" alt="{\displaystyle \Gamma (z)}" loading="lazy"></span> is the well known <a href="Gamma_function" title="Gamma function">gamma function</a> defined by

</p>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-z}\,dz,\quad \Re (z)>0}</annotation>

</semantics>

</math></span><img src="./cfa8f95810d521821a4b6c62728b1c0bd4425a63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.611ex; height:5.843ex;" alt="{\displaystyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-z}\,dz,\quad \Re (z)>0}" loading="lazy"></span>.</dd></dl></dd></dl>

<p>In the following it will be assumed that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta }</annotation>

</semantics>

</math></span><img src="./7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

</semantics>

</math></span><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"></span> are all complex numbers.

</p>

<div class="mw-heading mw-heading3"><h3 id="One-parameter_Mittag-Leffler_function">One-parameter Mittag-Leffler function</h3></div>

<p>The one-parameter Mittag-Leffler function is defined as<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{\alpha }(z)=\sum _{n=0}^{\infty }{\dfrac {z^{n}}{\Gamma (\alpha n+1)}}.}</annotation>

</semantics>

</math></span><img src="./ec5c99f8aee6b028fc3e327daa9a8a91d8090a68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.368ex; height:6.843ex;" alt="{\displaystyle E_{\alpha }(z)=\sum _{n=0}^{\infty }{\dfrac {z^{n}}{\Gamma (\alpha n+1)}}.}" loading="lazy"></span></dd></dl></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Two-parameter_Mittag-Leffler_function">Two-parameter Mittag-Leffler function</h3></div>

<p>The two-parameter Mittag-Leffler function is defined as<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{\alpha ,\beta }(z)=\sum _{n=0}^{\infty }{\dfrac {z^{n}}{\Gamma (\alpha n+\beta )}},\quad \Re (\alpha )>0.}</annotation>

</semantics>

</math></span><img src="./4c1e82fa1b2d1a6f2c73a2e282476ff302a32eff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.775ex; height:6.843ex;" alt="{\displaystyle E_{\alpha ,\beta }(z)=\sum _{n=0}^{\infty }{\dfrac {z^{n}}{\Gamma (\alpha n+\beta )}},\quad \Re (\alpha )>0.}" loading="lazy"></span></dd></dl></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Three-parameter_Mittag-Leffler_function_(Prabhakar_function)">Three-parameter Mittag-Leffler function (Prabhakar function)</h3></div>

<p>The three-parameter Mittag-Leffler function (Prabhakar function) is defined by<sup id="cite_ref-Tilak_1-1" class="reference"><a href="#cite_note-Tilak-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{\alpha ,\beta }^{\gamma }(z)=\sum _{n=0}^{\infty }{\dfrac {(\gamma )_{n}}{n!\Gamma (\alpha n+\beta )}}z^{n},\quad \Re (\alpha )>0}</annotation>

</semantics>

</math></span><img src="./46697794778685ecb3551749a302a003a8e42228.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:42.479ex; height:6.843ex;" alt="{\displaystyle E_{\alpha ,\beta }^{\gamma }(z)=\sum _{n=0}^{\infty }{\dfrac {(\gamma )_{n}}{n!\Gamma (\alpha n+\beta )}}z^{n},\quad \Re (\alpha )>0}" loading="lazy"></span></dd></dl></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\gamma )_{n}=\gamma (\gamma +1)\ldots (\gamma +n-1)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\gamma )_{n}=\gamma (\gamma +1)\ldots (\gamma +n-1)}</annotation>

</semantics>

</math></span><img src="./ffa5d5eddbf2736efa548a0e9d47e2b37f227eed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.532ex; height:2.843ex;" alt="{\displaystyle (\gamma )_{n}=\gamma (\gamma +1)\ldots (\gamma +n-1)}" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Elementary_special_cases">Elementary special cases</h2></div>

<p>The following special cases immediately follow from the definition.<sup id="cite_ref-Andrea_2-1" class="reference"><a href="#cite_note-Andrea-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p>

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{\alpha ,\beta }^{0}(z)={\frac {1}{\Gamma (\beta )}}}</annotation>

</semantics>

</math></span><img src="./cd116d5d864147f6a8b1e78132bdb959a284e421.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.825ex; height:6.009ex;" alt="{\displaystyle E_{\alpha ,\beta }^{0}(z)={\frac {1}{\Gamma (\beta )}}}" loading="lazy"></span></li>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{\alpha ,\beta }^{1}(z)=E_{\alpha ,\beta }(z)}</annotation>

</semantics>

</math></span><img src="./07d8099a5d87c6d71789a24c3cc0e04a7ee15b36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:17.691ex; height:3.509ex;" alt="{\displaystyle E_{\alpha ,\beta }^{1}(z)=E_{\alpha ,\beta }(z)}" loading="lazy"></span>, the two-parameter Mittag-Leffler function.</li>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{\alpha ,1}^{1}(z)=E_{\alpha }(z)}</annotation>

</semantics>

</math></span><img src="./77227fc718f3a9cc263d85f7312809972d1de93f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:16.171ex; height:3.509ex;" alt="{\displaystyle E_{\alpha ,1}^{1}(z)=E_{\alpha }(z)}" loading="lazy"></span>, the one-parameter Mittag-Leffler function.</li>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{1,1}^{1}(z)=e^{z}}</annotation>

</semantics>

</math></span><img src="./9e219459b6cd6e9d8c3690ccf4913bc3c8bcc69b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.13ex; height:3.509ex;" alt="{\displaystyle E_{1,1}^{1}(z)=e^{z}}" loading="lazy"></span>, the classical exponential function.</li></ol>

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Reduction_formula">Reduction formula</h3></div>

<p>The following formula can be reduced to lower the value of the third parameter <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

</semantics>

</math></span><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"></span>.<sup id="cite_ref-Andrea_2-2" class="reference"><a href="#cite_note-Andrea-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p>

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{\alpha ,\beta }^{\gamma +1}(z)={\frac {1}{\alpha \gamma }}{\big [}E_{\alpha ,\beta -1}^{\gamma }(z)+(1-\beta +\alpha \gamma )E_{\alpha ,\beta }^{\gamma }(z){\big ]}}</annotation>

</semantics>

</math></span><img src="./9802b106c25664305df306d3b3915d67b8699ef6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:48.808ex; height:5.676ex;" alt="{\displaystyle E_{\alpha ,\beta }^{\gamma +1}(z)={\frac {1}{\alpha \gamma }}{\big [}E_{\alpha ,\beta -1}^{\gamma }(z)+(1-\beta +\alpha \gamma )E_{\alpha ,\beta }^{\gamma }(z){\big ]}}" loading="lazy"></span></dd></dl></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Relation_with_Fox–Wright_function">Relation with Fox–Wright function</h3></div>

<p>The Prabhakar function is related to the <a href="Fox%E2%80%93Wright_function" title="Fox–Wright function">Fox–Wright function</a> by the following relation:

</p>

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mtr>

<mtd>

<mrow>

<mrow>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{\alpha ,\beta }^{\gamma }(z)={\frac {1}{\Gamma (\gamma )}}{}_{1}\Psi _{1}\left({\begin{matrix}\left(\gamma ,1\right)\\(\beta ,\alpha )\end{matrix}};z\right)}</annotation>

</semantics>

</math></span><img src="./b92e29f9f83d4723e077fb692d56663b86a26118.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.016ex; height:6.343ex;" alt="{\displaystyle E_{\alpha ,\beta }^{\gamma }(z)={\frac {1}{\Gamma (\gamma )}}{}_{1}\Psi _{1}\left({\begin{matrix}\left(\gamma ,1\right)\\(\beta ,\alpha )\end{matrix}};z\right)}" loading="lazy"></span></dd></dl></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Derivatives">Derivatives</h3></div>

<p>The derivative of the Prabhakar function is given by

</p>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dz}}\left(E_{\alpha ,\beta }^{\gamma }(z)\right)={\frac {1}{\alpha z}}{\big [}E_{\alpha ,\beta -1}^{\gamma }(z)+(1-\beta )E_{\alpha ,\beta }^{\gamma }{\big ]}}</annotation>

</semantics>

</math></span><img src="./3719dba01f7ff5122eed40f846db65d918b60138.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:45.828ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dz}}\left(E_{\alpha ,\beta }^{\gamma }(z)\right)={\frac {1}{\alpha z}}{\big [}E_{\alpha ,\beta -1}^{\gamma }(z)+(1-\beta )E_{\alpha ,\beta }^{\gamma }{\big ]}}" loading="lazy"></span></dd></dl></dd></dl>

<p>There is a general expression for higher order derivatives. Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math></span><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"></span> be a positive integer. The <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math></span><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"></span>-th derivative of the Prabhakar function is given by

</p>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {d^{m}}{dz^{m}}}\left(E_{\alpha ,\beta }^{\gamma }(z)\right)={\frac {\Gamma (\gamma +m)}{\Gamma (\gamma )}}E_{\alpha ,m\alpha +\beta }^{\gamma +m}(z)}</annotation>

</semantics>

</math></span><img src="./d231856fe23fc5d0f674701c1bc893f53bd151b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.685ex; height:6.509ex;" alt="{\displaystyle {\frac {d^{m}}{dz^{m}}}\left(E_{\alpha ,\beta }^{\gamma }(z)\right)={\frac {\Gamma (\gamma +m)}{\Gamma (\gamma )}}E_{\alpha ,m\alpha +\beta }^{\gamma +m}(z)}" loading="lazy"></span></dd></dl></dd></dl>

<p>The following result is useful in applications.

</p>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {d^{m}}{dz^{m}}}\left(t^{\beta -1}E_{\alpha ,\beta }^{\gamma }(t^{\alpha }z)\right)=t^{\beta -m-1}E_{\alpha ,\beta -m}^{\gamma }(t^{\alpha }z)}</annotation>

</semantics>

</math></span><img src="./6dda9d9394299213df40480b792b27f12294b7b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:43.589ex; height:5.509ex;" alt="{\displaystyle {\frac {d^{m}}{dz^{m}}}\left(t^{\beta -1}E_{\alpha ,\beta }^{\gamma }(t^{\alpha }z)\right)=t^{\beta -m-1}E_{\alpha ,\beta -m}^{\gamma }(t^{\alpha }z)}" loading="lazy"></span></dd></dl></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Integrals">Integrals</h3></div>

<p>The following result involving Prabhakar function is known.

</p>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{t}\tau ^{\beta -1}E_{\alpha ,\beta }^{\gamma }(\tau ^{\alpha }z)=t^{\beta }E_{\alpha ,\beta +1}^{\gamma }(t^{\alpha }z)}</annotation>

</semantics>

</math></span><img src="./491c8c694b68ddbf5d6ae2b883fd76721be5b65a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.68ex; height:6.176ex;" alt="{\displaystyle \int _{0}^{t}\tau ^{\beta -1}E_{\alpha ,\beta }^{\gamma }(\tau ^{\alpha }z)=t^{\beta }E_{\alpha ,\beta +1}^{\gamma }(t^{\alpha }z)}" loading="lazy"></span></dd></dl></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Laplace_transforms">Laplace transforms</h3></div>

<p>The following result involving <a href="Laplace_transform" title="Laplace transform">Laplace transforms</a> plays an important role in both physical applications and numerical computations of the Prabhakar function.

</p>

<dl><dd><dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L\left[t^{\beta -1}E_{\alpha ,\beta }^{\gamma }(t^{\alpha }z)\,;\,s\right]={\frac {s^{\alpha \gamma -\beta }}{(s^{\alpha }-z)^{\gamma }}},\quad \Re (s)>0,\quad |s|>|z|^{1/\alpha }}">

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L\left[t^{\beta -1}E_{\alpha ,\beta }^{\gamma }(t^{\alpha }z)\,;\,s\right]={\frac {s^{\alpha \gamma -\beta }}{(s^{\alpha }-z)^{\gamma }}},\quad \Re (s)>0,\quad |s|>|z|^{1/\alpha }}</annotation>

</semantics>

</math></span><img src="./1e240d3462a7b403ef214dfa8c509798e161b9d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:60.36ex; height:6.509ex;" alt="{\displaystyle L\left[t^{\beta -1}E_{\alpha ,\beta }^{\gamma }(t^{\alpha }z)\,;\,s\right]={\frac {s^{\alpha \gamma -\beta }}{(s^{\alpha }-z)^{\gamma }}},\quad \Re (s)>0,\quad |s|>|z|^{1/\alpha }}" loading="lazy"></span></dd></dl></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Prabhakar_fractional_calculus">Prabhakar fractional calculus</h2></div>

<p>The following function is known as the Prabhakar kernel in the literature.<sup id="cite_ref-Andrea_2-3" class="reference"><a href="#cite_note-Andrea-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e_{\alpha ,\beta }^{\gamma }(t;\lambda )=t^{\beta -1}E_{\alpha ,\beta }^{\gamma }(\lambda t^{\alpha })}</annotation>

</semantics>

</math></span><img src="./1873b6707597786df54ae4ebfe8a833c850d05f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:25.705ex; height:3.676ex;" alt="{\displaystyle e_{\alpha ,\beta }^{\gamma }(t;\lambda )=t^{\beta -1}E_{\alpha ,\beta }^{\gamma }(\lambda t^{\alpha })}" loading="lazy"></span></dd></dl></dd></dl>

<p>Given any function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation>

</semantics>

</math></span><img src="./5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}" loading="lazy"></span>, the convolution of the Prabhakar kernel and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation>

</semantics>

</math></span><img src="./5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}" loading="lazy"></span> is called the Prabhakar fractional integral:

</p>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{t_{0}}^{t}(t-u)^{\beta -1}E_{\alpha ,\beta }^{\gamma }\left(\lambda (t-u)^{\alpha }\right)f(u)\,du}</annotation>

</semantics>

</math></span><img src="./4d68133e83732c8d1b1a3858e06fdbc34b360102.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.175ex; height:6.509ex;" alt="{\displaystyle \int _{t_{0}}^{t}(t-u)^{\beta -1}E_{\alpha ,\beta }^{\gamma }\left(\lambda (t-u)^{\alpha }\right)f(u)\,du}" loading="lazy"></span></dd></dl></dd></dl>

<p>Properties of the Prabhakar fractional integral have been extensively studied in the literature.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFTilak_Raj_Prabhakar1971" class="citation journal cs1">Tilak Raj Prabhakar (1971). <a rel="nofollow" class="external text" href="https://ynu.repo.nii.ac.jp/record/6514/files/YMJ_19_N1_1971_007-015.pdf">"A singular integral equation with a generalized Mittag-Leffler function in the kernel"</a> <span class="cs1-format">(PDF)</span>. <i>Yoklohama Mathematics Journal</i>. <b>19</b> (1): <span class="nowrap">7–</span>15<span class="reference-accessdate">. Retrieved <span class="nowrap">27 December</span> 2023</span>.</cite></span>

</li>

<li id="cite_note-Andrea-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Andrea_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Andrea_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Andrea_2-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Andrea_2-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFAndrea_Giusti,_Ivano_Colombaro,_Roberto_Garra2020" class="citation journal cs1">Andrea Giusti, Ivano Colombaro, Roberto Garra (2020). "A practical guide to Prabhakar fractional calculus". <i>Fractional Calculus and Applied Analysis</i>. <b>25</b> (1): <span class="nowrap">9–</span>54. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2002.10978">2002.10978</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2Ffca-2020-0002">10.1515/fca-2020-0002</a>.</cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{cite journal}}</code>: CS1 maint: multiple names: authors list (link)</span></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFRudolf_Gorenflo,_Anatoly_A._Kilbas,_Francesco_Mainardi,_Sergei_V._Rogosin2014" class="citation book cs1">Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergei V. Rogosin (2014). <i>Mittag-Leffler Functions, Related Topics and Applications</i>. Springer. p. 17. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-662-43929-6</bdi>.</cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{cite book}}</code>: CS1 maint: multiple names: authors list (link)</span></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFRudolf_Gorenflo,_Anatoly_A._Kilbas,_Francesco_Mainardi,_Sergei_V._Rogosin2014" class="citation book cs1">Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergei V. Rogosin (2014). <i>Mittag-Leffler Functions, Related Topics and Applications</i>. Springer. p. 56. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-662-43929-6</bdi>.</cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{cite book}}</code>: CS1 maint: multiple names: authors list (link)</span></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFRudolf_Gorenflo,_Anatoly_A._Kilbas,_Francesco_Mainardi,_Sergei_V._Rogosin2014" class="citation book cs1">Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergei V. Rogosin (2014). <i>Mittag-Leffler Functions, Related Topics and Applications</i>. Springer. p. 97. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-662-43929-6</bdi>.</cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{cite book}}</code>: CS1 maint: multiple names: authors list (link)</span></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFRoberto_Garra_and_Roberto_Garrappa2018" class="citation journal cs1">Roberto Garra and Roberto Garrappa (2018). "The Prabhakar or three parameter Mittag–Leffler function: theory and application". <i>Communications in Nonlinear Science and Numerical Simulation</i>. <b>56</b>: <span class="nowrap">314–</span>329. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1708.07298v2">1708.07298v2</a></span>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2018CNSNS..56..314G">2018CNSNS..56..314G</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.cnsns.2017.08.018">10.1016/j.cnsns.2017.08.018</a>.</cite></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFAnatoly_A._Kilbas,_Megumi_Saigo_and_R._K._Saxena2004" class="citation journal cs1">Anatoly A. Kilbas, Megumi Saigo and R. K. Saxena (2004). "Generalized mittag-leffler function and generalized fractional calculus operators". <i>Integral Transforms and Special Functions</i>. <b>15</b> (1): <span class="nowrap">31–</span>49. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F10652460310001600717">10.1080/10652460310001600717</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120569191">120569191</a>.</cite></span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFF._Polito_and_Z._Tomovski2016" class="citation journal cs1">F. Polito and Z. Tomovski (2016). "Some properties of Prabhakar-type fractional calculus operators". <i>Fractional Differential Calculus</i>. <b>6</b> (1): <span class="nowrap">73–</span>94. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1508.03224">1508.03224</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.7153%2Ffdc-06-05">10.7153/fdc-06-05</a>.</cite></span>

</li>

</ol></div></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}

/* end https://en.wikipedia.org/ */

</style></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-04-21" href="https://en.wikipedia.org/wiki/?title=Prabhakar_function&oldid=1286736416">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>