imported>Arisu8899: /* growthexperiments-addlink-summary-summary:3|0|0 */

2025-11-01T14:29:10Z

growthexperiments-addlink-summary-summary:3|0|0

← Previous revision		Revision as of 14:29, 1 November 2025
Line 39:		Line 39:
	:<math>\begin{align}		:<math>\begin{align}
	(a)_0 &= 1, \\		(a)_0 &= 1, \\
	(a)_n &= a(a+1)(a+2) \cdots (a+n-1), && n \geq 1		(a)_n &= a(a+1)(a+2) \cdots (a+n-1) = \frac{\Gamma(a+n)}{\Gamma(a)}, && n \geq 1,
	\end{align}</math>		\end{align}</math>

	this can be written		where <math>\Gamma(x)</math> represents the [[Gamma function\|gamma function]], this can be written

	:<math>\,{}_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z) = \sum_{n=0}^\infty \frac{(a_1)_n\cdots(a_p)_n}{(b_1)_n\cdots(b_q)_n} \, \frac {z^n} {n!}.</math>		:<math>\,{}_pF_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z) = \sum_{n=0}^\infty \frac{(a_1)_n\cdots(a_p)_n}{(b_1)_n\cdots(b_q)_n} \, \frac {z^n} {n!} = \frac{\Gamma(b_1)\cdots \Gamma(b_q)}{\Gamma(a_1)\cdots \Gamma(a_p)} \sum_{n=0}^{\infty} \frac{\Gamma(n+a_1)\cdots \Gamma(n+a_p)}{\Gamma(n+b_1)\cdots \Gamma(n+b_q)} \frac{z^n}{n!}.</math>

	(Note that this use of the Pochhammer symbol is not standard; however it is the standard usage in this context.)		(Note that this use of the Pochhammer symbol is not standard; however it is the standard usage in this context.)
Line 65:		Line 65:

	Excluding these cases, the [[ratio test]] can be applied to determine the radius of convergence.		Excluding these cases, the [[ratio test]] can be applied to determine the radius of convergence.
	* If ''p'' < ''q'' + 1 then the ratio of coefficients tends to zero. This implies that the series converges for any finite value of ''z'' and thus defines an entire function of ''z''. An example is the power series for the exponential function.		* If ''p'' < ''q'' + 1 then the ratio of coefficients tends to zero. This implies that the series converges for any finite value of ''z'' and thus defines an [[entire function]] of ''z''. An example is the power series for the exponential function.
	* If ''p'' = ''q'' + 1 then the ratio of coefficients tends to one. This implies that the series converges for \|''z''\| < 1 and diverges for \|''z''\| > 1. Whether it converges for \|''z''\| = 1 is more difficult to determine. Analytic continuation can be employed for larger values of ''z''.		* If ''p'' = ''q'' + 1 then the ratio of coefficients tends to one. This implies that the series converges for \|''z''\| < 1 and diverges for \|''z''\| > 1. Whether it converges for \|''z''\| = 1 is more difficult to determine. Analytic continuation can be employed for larger values of ''z''.
	* If ''p'' > ''q'' + 1 then the ratio of coefficients grows without bound. This implies that, besides ''z'' = 0, the series diverges. This is then a divergent or asymptotic series, or it can be interpreted as a symbolic shorthand for a differential equation that the sum satisfies formally.		* If ''p'' > ''q'' + 1 then the ratio of coefficients grows without bound. This implies that, besides ''z'' = 0, the series diverges. This is then a divergent or asymptotic series, or it can be interpreted as a symbolic shorthand for a differential equation that the sum satisfies formally.

	The question of convergence for ''p''=''q''+1 when ''z'' is on the unit circle is more difficult. It can be shown that the series converges absolutely at ''z'' = 1 if		The question of convergence for ''p''=''q''+1 when ''z'' is on the [[unit circle]] is more difficult. It can be shown that the series converges absolutely at ''z'' = 1 if
	:<math>\Re\left(\sum b_k - \sum a_j\right)>0</math>.		:<math>\Re\left(\sum b_k - \sum a_j\right)>0</math>.
	Further, if ''p''=''q''+1, <math>\sum_{i=1}^{p}a_{i}\geq\sum_{j=1}^{q}b_{j}</math> and ''z'' is real, then the following convergence result holds {{harvtxt\|Quigley\|Wilson\|Walls\|Bedford\|2013}}:		Further, if ''p''=''q''+1, <math>\sum_{i=1}^{p}a_{i}\geq\sum_{j=1}^{q}b_{j}</math> and ''z'' is real, then the following convergence result holds {{harvtxt\|Quigley\|Wilson\|Walls\|Bedford\|2013}}:
Line 215:		Line 215:
	{{Main\|Dixon's identity}}		{{Main\|Dixon's identity}}

	Dixon's identity,<ref>See {{harv\|Bailey\|1935\|loc=Section 3.1}} for a detailed proof. An alternative proof is in {{harv\|Slater\|1966\|loc=Section 2.3.3}}</ref> first proved by {{harvtxt\|Dixon\|1902}}, gives the sum of a well-poised <sub>3</sub>''F''<sub>2</sub> at 1:		[[Dixon's identity]],<ref>See {{harv\|Bailey\|1935\|loc=Section 3.1}} for a detailed proof. An alternative proof is in {{harv\|Slater\|1966\|loc=Section 2.3.3}}</ref> first proved by {{harvtxt\|Dixon\|1902}}, gives the sum of a well-poised <sub>3</sub>''F''<sub>2</sub> at 1:
	:<math>{}_3F_2 (a,b,c;1+a-b,1+a-c;1)= \frac{\Gamma(1+\frac{a}{2})\Gamma(1+\frac{a}{2}-b-c)\Gamma(1+a-b)\Gamma(1+a-c)}{\Gamma(1+a)\Gamma(1+a-b-c)\Gamma(1+\frac{a}{2}-b)\Gamma(1+\frac{a}{2}-c)}.</math>		:<math>{}_3F_2 (a,b,c;1+a-b,1+a-c;1)= \frac{\Gamma(1+\frac{a}{2})\Gamma(1+\frac{a}{2}-b-c)\Gamma(1+a-b)\Gamma(1+a-c)}{\Gamma(1+a)\Gamma(1+a-b-c)\Gamma(1+\frac{a}{2}-b)\Gamma(1+\frac{a}{2}-c)}.</math>
	For generalization of Dixon's identity, see a paper by Lavoie, et al.		For generalization of Dixon's identity, see a paper by Lavoie, et al.
Line 336:		Line 336:
	Relations to other functions are known for certain parameter combinations only.		Relations to other functions are known for certain parameter combinations only.

	The function <math>x\; {}_1F_2\left(\frac{1}{2};\frac{3}{2},\frac{3}{2};-\frac{x^2}{4}\right)</math> is the antiderivative of the [[cardinal sine]]. With modified values of <math>a_1</math> and <math>b_1</math>, one obtains the antiderivative of <math>\sin(x^\beta)/x^\alpha</math>.<ref>Victor Nijimbere, Ural Math J vol 3(1) and https://arxiv.org/abs/1703.01907 (2017)</ref>		The function <math>x\; {}_1F_2\left(\frac{1}{2};\frac{3}{2},\frac{3}{2};-\frac{x^2}{4}\right)</math> is the [[antiderivative]] of the [[cardinal sine]]. With modified values of <math>a_1</math> and <math>b_1</math>, one obtains the antiderivative of <math>\sin(x^\beta)/x^\alpha</math>.<ref>Victor Nijimbere, Ural Math J vol 3(1) and https://arxiv.org/abs/1703.01907 (2017)</ref>

	The [[Lommel function]] is <math> s_{\mu, \nu} (z) = \frac{z^{\mu + 1}}{(\mu - \nu + 1)(\mu + \nu + 1)} {}_1F_2(1; \frac{\mu}{2} - \frac{\nu}{2} + \frac{3}{2} , \frac{\mu}{2} + \frac{\nu}{2} + \frac{3}{2} ;-\frac{z^2}{4}) </math>.<ref>Watson's "Treatise on the Theory of Bessel functions" (1966), Section 10.7, Equation (10)</ref>		The [[Lommel function]] is <math> s_{\mu, \nu} (z) = \frac{z^{\mu + 1}}{(\mu - \nu + 1)(\mu + \nu + 1)} {}_1F_2\left(1; \frac{\mu}{2} - \frac{\nu}{2} + \frac{3}{2} , \frac{\mu}{2} + \frac{\nu}{2} + \frac{3}{2} ;-\frac{z^2}{4}\right) </math>.<ref>Watson's "Treatise on the Theory of Bessel functions" (1966), Section 10.7, Equation (10)</ref>

	===The series <sub>2</sub>''F''<sub>0</sub>===		===The series <sub>2</sub>''F''<sub>0</sub>===
Line 379:		Line 379:

	:<math>\int_0^x\sqrt{1+y^\alpha}\,\mathrm{d}y=\frac{x}{2+\alpha}\left \{\alpha\;{}_2F_1\left(\tfrac{1}{\alpha},\tfrac{1}{2};1+\tfrac{1}{\alpha};-x^\alpha \right) +2\sqrt{x^\alpha+1} \right \},\qquad \alpha\neq0.</math>		:<math>\int_0^x\sqrt{1+y^\alpha}\,\mathrm{d}y=\frac{x}{2+\alpha}\left \{\alpha\;{}_2F_1\left(\tfrac{1}{\alpha},\tfrac{1}{2};1+\tfrac{1}{\alpha};-x^\alpha \right) +2\sqrt{x^\alpha+1} \right \},\qquad \alpha\neq0.</math>

			===The series <sub>2</sub>''F''<sub>2</sub>===
			The hypergeometric series <math>{}_2F_2</math> is generally associated with integrals of products of power functions and the exponential function. As such, the [[Exponential integral\|exponential integral]] can be written as:
			:<math>\operatorname{Ei}(x)=x{}_2F_2(1,1;2,2;x)+\ln x+\gamma.</math>

	===The series <sub>3</sub>''F''<sub>0</sub>===		===The series <sub>3</sub>''F''<sub>0</sub>===
Line 390:		Line 394:
	is the [[dilogarithm]]<ref>{{cite web\|last=Candan\|first=Cagatay\|title=A Simple Proof of F(1,1,1;2,2;x)=dilog(1-x)/x		is the [[dilogarithm]]<ref>{{cite web\|last=Candan\|first=Cagatay\|title=A Simple Proof of F(1,1,1;2,2;x)=dilog(1-x)/x
	\|url=http://www.eee.metu.edu.tr/~ccandan/pub_dir/hyper_rel.pdf}}</ref>		\|url=http://www.eee.metu.edu.tr/~ccandan/pub_dir/hyper_rel.pdf}}</ref>

			Furthermore,
			::<math>_3F_2(1,1,1+n;2,2;x)=\frac{1}{n!}\sum^{n}_{k=0} \biggl[{n \atop k}\biggr] \frac{\operatorname{Li}_{2-k}(x)}{x}</math>,
			where <math>\biggl[{n \atop k}\biggr]</math> is the unsigned [[Stirling number of the first kind]].
			<ref>{{Cite journal \|last=Angervuori \|first=Ilari \|last2=Haenggi \|first2=Martin \|last3=Wichman \|first3=Risto \|title=Meta Distribution of the SIR in a Narrow-Beam LEO Uplink \|url=https://ieeexplore.ieee.org/document/10909705 \|journal=[[IEEE Transactions on Communications]] \|volume=73 \|issue=9}}</ref>

	The function		The function
Line 462:		Line 471:
	* {{cite book \| last= Bailey \| first= W.N. \| title= Generalized Hypergeometric Series \| publisher= Cambridge University Press \| location= London \| year= 1935 \| series= Cambridge Tracts in Mathematics and Mathematical Physics \| volume= 32 \| zbl= 0011.02303 }}		* {{cite book \| last= Bailey \| first= W.N. \| title= Generalized Hypergeometric Series \| publisher= Cambridge University Press \| location= London \| year= 1935 \| series= Cambridge Tracts in Mathematics and Mathematical Physics \| volume= 32 \| zbl= 0011.02303 }}
	* {{cite journal \| last= Dixon \| first= A.C. \| title= Summation of a certain series \| journal= Proc. London Math. Soc. \| year= 1902 \| volume= 35 \| issue= 1 \| pages= 284–291 \| doi=10.1112/plms/s1-35.1.284 \| jfm= 34.0490.02 \| url= https://zenodo.org/record/1433433 }}		* {{cite journal \| last= Dixon \| first= A.C. \| title= Summation of a certain series \| journal= Proc. London Math. Soc. \| year= 1902 \| volume= 35 \| issue= 1 \| pages= 284–291 \| doi=10.1112/plms/s1-35.1.284 \| jfm= 34.0490.02 \| url= https://zenodo.org/record/1433433 }}
	* {{cite journal \| last= Dougall \| first= J. \| title= On Vandermonde's theorem and some more general expansions \| journal= Proc. Edinburgh Math. Soc. \| year= 1907 \| volume= 25 \| pages= 114–132 \| doi= 10.1017/S0013091500033642 ~~\| doi-broken-date= 20 December 2024~~ \| doi-access= free }}		* {{cite journal \| last= Dougall \| first= J. \| title= On Vandermonde's theorem and some more general expansions \| journal= Proc. Edinburgh Math. Soc. \| year= 1907 \| volume= 25 \| pages= 114–132 \| doi= 10.1017/S0013091500033642 \| doi-access= free }}
	*{{Cite book \| last1=Erdélyi \| first1=Arthur \| last2=Magnus \| first2=Wilhelm \| author2-link=Wilhelm Magnus \| last3=Oberhettinger \| first3=Fritz \| last4=Tricomi \| first4=Francesco G. \| title=Higher transcendental functions. Vol. III \| publisher=McGraw-Hill Book Company, Inc., New York-Toronto-London \| mr=0066496 \| year=1955}}		*{{Cite book \| last1=Erdélyi \| first1=Arthur \| last2=Magnus \| first2=Wilhelm \| author2-link=Wilhelm Magnus \| last3=Oberhettinger \| first3=Fritz \| last4=Tricomi \| first4=Francesco G. \| title=Higher transcendental functions. Vol. III \| publisher=McGraw-Hill Book Company, Inc., New York-Toronto-London \| mr=0066496 \| year=1955}}
	* {{cite book \| last1= Gasper \| first1= George \| last2= Rahman \| first2= Mizan \| author-link2= Mizan Rahman \| title= Basic Hypergeometric Series \| edition= 2nd \| year= 2004 \| series= Encyclopedia of Mathematics and Its Applications \| volume= 96 \| publisher= Cambridge University Press \| location= Cambridge, UK \| mr= 2128719 \| zbl= 1129.33005 \| isbn= 978-0-521-83357-8 }} (the first edition has {{isbn\|0-521-35049-2}})		* {{cite book \| last1= Gasper \| first1= George \| last2= Rahman \| first2= Mizan \| author-link2= Mizan Rahman \| title= Basic Hypergeometric Series \| edition= 2nd \| year= 2004 \| series= Encyclopedia of Mathematics and Its Applications \| volume= 96 \| publisher= Cambridge University Press \| location= Cambridge, UK \| mr= 2128719 \| zbl= 1129.33005 \| isbn= 978-0-521-83357-8 }} (the first edition has {{isbn\|0-521-35049-2}})

imported>Citation bot: Altered template type. Add: bibcode, arxiv, doi, page, issue, volume, journal. Removed parameters. Some additions/deletions were parameter name changes. | Use this bot. Report bugs. | Suggested by Headbomb | Category:Articles with missing Cite arXiv inputs | #UCB_Category 12/25

2025-06-19T16:16:08Z

Altered template type. Add: bibcode, arxiv, doi, page, issue, volume, journal. Removed parameters. Some additions/deletions were parameter name changes. | Use this bot. Report bugs. | Suggested by Headbomb | Category:Articles with missing Cite arXiv inputs | #UCB_Category 12/25

@@ Line 410: / Line 410: @@
 </ref>
 ::<math>\operatorname{BR}(a) = -a \; {}_4F_3\left( \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5} ; \frac{1}{2}, \frac{3}{4}, \frac{5}{4} ; \frac{3125a^4}{256} \right).</math>
 ===The series <sub>q+1</sub>''F''<sub>q</sub>===

imported>JJMC89 bot III: Moving :Category:Mathematical series to :Category:Series (mathematics) per Wikipedia:Categories for discussion/Speedy

2025-04-14T21:16:59Z

Moving Category:Mathematical series to Category:Series (mathematics) per Wikipedia:Categories for discussion/Speedy

Show changes

Generalized hypergeometric function - Revision history

imported>Arisu8899: /* growthexperiments-addlink-summary-summary:3|0|0 */

imported>JJMC89 bot III: Moving :Category:Mathematical series to :Category:Series (mathematics) per Wikipedia:Categories for discussion/Speedy