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Hypergeometric function

The hypergeometric function ( Gauss function ) is defined inside the circle|z|<one {\ displaystyle | z | <1} | z | <1 as the sum of the hypergeometric series

F(a,b;c;z)=one+∑k=one∞[∏l=0k-one(a+l)(b+l)(one+l)(c+l)]zk=one+abczone!+a(a+one)b(b+one)c(c+one)z22!+...,{\ displaystyle F (a, b; c; z) = 1 + \ sum _ {k = 1} ^ {\ infty} \ left [\ prod _ {l = 0} ^ {k-1} {(a + l) (b + l) \ over (1 + l) (c + l)} \ right] z ^ {k} = 1 + {\ frac {ab} {c}} {\ frac {z} {1! }} + {\ frac {a (a + 1) b (b + 1)} {c (c + 1)}} {\ frac {z ^ {2}} {2!}} + \ dots,} F (a, b; c; z) = 1 + \ sum _ {{k = 1}} ^ {\ infty} \ left [\ prod _ {{l = 0}} ^ {{k-1}} { (a + l) (b + l) \ over (1 + l) (c + l)} \ right] z ^ {k} = 1 + {\ frac {ab} {c}} {\ frac {z} {1!}} + {\ Frac {a (a + 1) b (b + 1)} {c (c + 1)}} {\ frac {z ^ {2}} {2!}} + \ Dots ,

and when|z|>one {\ displaystyle | z |> 1} | z |> 1 - as its analytical continuation . It is a solution to a linear ordinary differential equation (ODE) of the second order, called the hypergeometric equation.

Content

  • 1 History
  • 2 Hypergeometric equation
  • 3 particular values ​​forz=one/2 {\ displaystyle z = 1/2} {\ displaystyle z = 1/2}
  • 4 Recording other functions through hypergeometric
    • 4.1 Examples
  • 5 Identities
  • 6 notes
  • 7 Literature

History

The term "hypergeometric series" was first used by John Wallis in 1655 in the book Arithmetica Infinitorum . This term refers to a series whose general formula of members has the form [1]

one⋅3⋅5⋅...⋅(2n+one)2⋅four⋅...⋅2n.{\ displaystyle {\ frac {1 \ cdot 3 \ cdot 5 \ cdot \ ldots \ cdot (2n + 1)} {2 \ cdot 4 \ cdot \ ldots \ cdot 2n}}.} {\frac  {1\cdot 3\cdot 5\cdot \ldots \cdot (2n+1)}{2\cdot 4\cdot \ldots \cdot 2n}}.

Hypergeometric series were studied by Leonard Euler , and in more detail by Gauss [2] . In the 19th century, the study was continued by Ernst Kummer, and Bernhard Riemann defined a hypergeometric function through an equation that it satisfies.

Hypergeometric equation

Consider the Euler differential equationz(one-z)d2udz2+[c-(a+b+one)z]dudz-abu=0, {\ displaystyle z (1-z) {\ frac {d ^ {2} u} {dz ^ {2}}} + [c- (a + b + 1) z] {\ frac {du} {dz} } -abu = 0,} z(1-z){\frac  {d^{2}u}{dz^{2}}}+[c-(a+b+1)z]{\frac  {du}{dz}}-abu=0, where the parameters a , b, and c can be arbitrary complex numbers. Its generalization to arbitrary regular singular points is given by the Riemann differential equation . The Euler equation has three singular points : 0, 1 and∞ {\ displaystyle \ infty} \infty .

When parameterc {\ displaystyle c} c non-zero and negative integers(c≠0,-one,-2,...) {\ displaystyle (c \ neq 0, -1, -2, \ ldots)} (c\neq 0,-1,-2,\ldots ) a regular solution to zero of the Euler equation can be written in a series called hypergeometric:

2Fone(a,b;c;z)≡F(a,b;c;z)=one+abczone!+a(a+one)b(b+one)c(c+one)z22!+....{\ displaystyle _ {2} F_ {1} (a, b; c; z) \ equiv F (a, b; c; z) = 1 + {\ frac {ab} {c}} {\ frac {z } {1!}} + {\ Frac {a (a + 1) b (b + 1)} {c (c + 1)}} {\ frac {z ^ {2}} {2!}} + \ dots.} _{2}F_{1}(a,b;c;z)\equiv F(a,b;c;z)=1+{\frac  {ab}{c}}{\frac  {z}{1!}}+{\frac  {a(a+1)b(b+1)}{c(c+1)}}{\frac  {z^{2}}{2!}}+\dots .

This function is called hypergeometric. Often the designation is used ( Pohhammer symbol )

(p)n=Γ(p+n)Γ(p),{\ displaystyle (p) _ {n} = {\ frac {\ Gamma (p + n)} {\ Gamma (p)}},}  

WhereΓ {\ displaystyle \ Gamma}   - gamma function . Then the hypergeometric function can be represented as

F(a,b;c;z)=∑n=0∞(a)n(b)nzn(c)nn!.{\ displaystyle F (a, b; c; z) = \ sum _ {n = 0} ^ {\ infty} {\ frac {(a) _ {n} (b) _ {n} z ^ {n} } {(c) _ {n} n!}}.}  

Designation2Fone(a,b;c;z) {\ displaystyle _ {2} F_ {1} (a, b; c; z)}   indicate that there are two parameters, a and b, "going to the numerator", and one, c, "going to the denominator." On the border|z|=one {\ displaystyle | z | = 1}   the series through which the hypergeometric function is determined absolutely converges if the real part of the suma+b-c<0 {\ displaystyle a + bc <0}   conditionally converges atz≠one {\ displaystyle z \ neq 1}   ,0≤a+b-c<one {\ displaystyle 0 \ leq a + bc <1}   and diverges ifa+b-c≥one {\ displaystyle a + bc \ geq 1}   . The second linearly independent solution of the Euler differential equation has the form

zone-cF(b-c+one,a-c+one;2-c;z){\ displaystyle \ z ^ {1-c} F (b-c + 1, a-c + 1; 2-c; z)}  

It has a singular point forz=0 {\ displaystyle z = 0}   and fair for all non-positivec {\ displaystyle c}  (c=0,-one,-2,...) {\ displaystyle (c = 0, -1, -2, \ ldots)}   . [3]

Integral representation for hypergeometric function forRe(c)>Re(b)>0 {\ displaystyle {\ text {Re}} (c)> {\ text {Re}} (b)> 0}   (Euler's formula) can be written as follows:

F(a,b;c;z)=Γ(c)Γ(b)Γ(c-b)∫0onetb-one(one-t)c-b-one(one-tz)-adt,{\ displaystyle F (a, b; c; z) = {\ Gamma (c) \ over \ Gamma (b) \ Gamma (cb)} \ int \ limits _ {0} ^ {1} t ^ {b- 1} (1-t) ^ {cb-1} (1-tz) ^ {- a} \, dt,}  

WhereΓ(x) {\ displaystyle \ Gamma (x)}   - gamma function of Euler. This expression is a unique analytic function on the complexz {\ displaystyle z}   planes with a cut along the real axis fromone {\ displaystyle 1}   before∞ {\ displaystyle \ infty}   and provides an analytic continuation to the entire complex plane for the hypergeometric series that converges only when|z|<one {\ displaystyle \ left | z \ right | <1}   .

Particular values ​​forz=one/2 {\ displaystyle z = 1/2} {\ displaystyle z = 1/2}

The second Gauss summation theorem is expressed by the formula:

2Fone(a,b;one2(one+a+b);one2)=Γ(one2)Γ(one2(one+a+b))Γ(one2(one+a))Γ(one2(one+b)).{\ displaystyle _ {2} F_ {1} \ left (a, b; {\ tfrac {1} {2}} \ left (1 + a + b \ right); {\ tfrac {1} {2}} \ right) = {\ frac {\ Gamma ({\ tfrac {1} {2}}) \ Gamma ({\ tfrac {1} {2}} \ left (1 + a + b \ right))} {\ Gamma ({\ tfrac {1} {2}} \ left (1 + a) \ right) \ Gamma ({\ tfrac {1} {2}} \ left (1 + b \ right))}}.}  

Bailey's theorem is expressed by the formula:

2Fone(a,one-a;c;one2)=Γ(one2c)Γ(one2(one+c))Γ(one2(c+a))Γ(one2(one+c-a)).{\ displaystyle _ {2} F_ {1} \ left (a, 1-a; c; {\ tfrac {1} {2}} \ right) = {\ frac {\ Gamma ({\ tfrac {1} { 2}} c) \ Gamma ({\ tfrac {1} {2}} \ left (1 + c \ right))} {\ Gamma ({\ tfrac {1} {2}} \ left (c + a \ right)) \ Gamma ({\ tfrac {1} {2}} \ left (1 + ca \ right))}}.}  

Recording other functions through hypergeometric

An important property of a hypergeometric function is that many special and elementary functions can be obtained from it for certain parameter values ​​and the transformation of an independent argument.

Examples

  • (one+x)n=F(-n,b;b;-x){\ displaystyle \ left (1 + x \ right) ^ {n} = F (-n, b; b; -x)}  
  • xn=F(-n,b;b;one-x){\ displaystyle x ^ {n} = F \ left (-n, b; b; 1-x \ right)}  
  • onexln⁡(one+x)=F(one,one;2;-x){\ displaystyle {1 \ over x} \ ln (1 + x) = F (1,1; 2; -x)}  
onexarcsin⁡(x)=F(one2,one2;32;x2){\ displaystyle {1 \ over x} \ arcsin (x) = F \ left ({\ frac {1} {2}}, {\ frac {1} {2}}; {\ frac {3} {2} }; x ^ {2} \ right)}  
  • ex=limn→∞F(one,n;one;xn){\ displaystyle e ^ {x} = \ lim _ {n \ to \ infty} F \ left (1, n; 1; {x \ over n} \ right)}  
  • cos⁡x=lima,b→∞F(a,b;one2;-x2fourab){\ displaystyle \ cos x = \ lim _ {a, \; b \ to \ infty} F \ left (a, b; {\ frac {1} {2}}; - {\ frac {x ^ {2} } {4ab}} \ right)}  
  • cosh⁡x=lima,b→∞F(a,b;one2;x2fourab){\ displaystyle \ cosh x = \ lim _ {a, \; b \ to \ infty} F \ left (a, b; {\ frac {1} {2}}; {x ^ {2} \ over 4ab} \ right)}  
  • Complete elliptic integral of the first kind:
    K(k)=∫0π/2dφone-k2sin2⁡φ=π2F(one2,one2;one;k2){\ displaystyle K (k) = \ int \ limits _ {0} ^ {\ pi / 2} \! {\ frac {d \ varphi} {\ sqrt {1-k ^ {2} \ sin ^ {2} \ varphi}}} = {\ frac {\ pi} {2}} F \ left ({\ frac {1} {2}}, {\ frac {1} {2}}; 1; k ^ {2} \ right)}  
  • The complete elliptic integral of the second kind:
    E(k)=∫0π/2one-k2sin2⁡φdφ=π2F(-one2,one2;one;k2){\ displaystyle E (k) = \ int \ limits _ {0} ^ {\ pi / 2} \! {\ sqrt {1-k ^ {2} \ sin ^ {2} \ varphi}} \, d \ varphi = {\ frac {\ pi} {2}} F \ left (- {\ frac {1} {2}}, {\ frac {1} {2}}; 1; k ^ {2} \ right) }  
  • Legendre Polynomial :
    Pn(x)=F(n+one,-n;one;one-x2){\ displaystyle P_ {n} (x) = F (n + 1, -n; 1; {\ frac {1-x} {2}})}  
  • The attached Legendre function :
    Pn,m(x)=(one-x2)m2Γ(n+m+one)2mΓ(n-m+one)Γ(m+one)F(n+m+one,m-n;m+one;one-x2){\ displaystyle P_ {n, \; m} (x) = (1-x ^ {2}) ^ {\ frac {m} {2}} {\ Gamma (n + m + 1) \ over 2 ^ { m} \ Gamma (n-m + 1) \ Gamma (m + 1)} F \ left (n + m + 1, mn; m + 1; {\ frac {1-x} {2}} \ right) }  
  • Bessel Functions :
    Jν(x)=lima,b→∞[(x2)νΓ(ν+one)F(a,b;ν+one;-x2fourab)]{\ displaystyle J _ {\ nu} (x) = \ lim _ {a, \; b \ to \ infty} \ left [{\ frac {\ left ({\ dfrac {x} {2}} \ right) ^ {\ nu}} {\ Gamma (\ nu +1)}} F \ left (a, b; \ nu +1; - {\ frac {x ^ {2}} {4ab}} \ right) \ right] }  
  • Kummer (Pohhammer) function , or degenerate hypergeometric function
    M(a,c,z)=oneFone(a,c,z)=limb→∞F(a,b;c;z/b){\ displaystyle M (a, c, z) = {} _ {1} F_ {1} (a, c, z) = \ lim _ {b \ to \ infty} F (a, b; c; z / b)}  
    is a solution to a degenerate hypergeometric equation
    zd2wdz2+(c-z)dwdz-aw=0.{\ displaystyle z {\ frac {d ^ {2} w} {dz ^ {2}}} + (cz) {\ frac {dw} {dz}} - aw = 0.}  
  • A degenerate hypergeometric function with an integer nonpositive first argument is a generalized Laguerre polynomial :
    Lnλ(x)=oneFone(-n,λ,x).{\ displaystyle L_ {n} ^ {\ lambda} (x) = {} _ {1} F_ {1} (- n, \ lambda, x).}  

Identities

  • 27(z-one)2⋅2Fone(onefour,3four;23;z)8+eighteen(z-one)⋅2Fone(onefour,3four;23;z)four-8⋅2Fone(onefour,3four;23;z)2=one{\ displaystyle 27 \, (z-1) ^ {2} \ cdot {_ {2} F_ {1}} \ left ({\ tfrac {1} {4}}, {\ tfrac {3} {4} }; {\ tfrac {2} {3}}; z \ right) ^ {8} +18 \, (z-1) \ cdot {_ {2} F_ {1}} \ left ({\ tfrac {1 } {4}}, {\ tfrac {3} {4}}; {\ tfrac {2} {3}}; z \ right) ^ {4} -8 \ cdot {_ {2} F_ {1}} \ left ({\ tfrac {1} {4}}, {\ tfrac {3} {4}}; {\ tfrac {2} {3}}; z \ right) ^ {2} = 1}  
  • And a wonderful special case of the previous expression:
    2Fone(onefour,3four;23;one3)=onefour2-four3+four3+four-2-four3-2{\ displaystyle _ {2} F_ {1} \ left ({\ frac {1} {4}}, {\ frac {3} {4}}; \, {\ frac {2} {3}}; \ , {\ frac {1} {3}} \ right) = {\ frac {1} {\ sqrt {{\ sqrt {{\ frac {4} {\ sqrt {2 - {\ sqrt [{3}] { 4}}}}} + {\ sqrt [{3}] {4}} + 4}} - {\ sqrt {2 - {\ sqrt [{3}] {4}}}} - 2}}}}  

Notes

  1. ↑ Scott, 1981 , p. 16.
  2. ↑ Vinogradov, 1977 , p. 1004.
  3. ↑ Bateman, Erdeyi, T. 1, 1973 , p. 69-70.

Literature

  • Mathematical Encyclopedia / Ed. I.M. Vinogradova. - M. , 1977 .-- T. 1.
  • Beitman G., Erdeyi A. Higher Transcendental Functions = Higher Transcendental Functions / Per. N. Ya. Vilenkina. - Ed. 2nd. - M .: Nauka, 1973. - T. 1. - 296 p. - 14,000 copies.
  • Kuznetsov D. S .: Special functions - M.: "Higher school", 1962
  • Landau, L.D. , Lifshits, E.M. Quantum mechanics (nonrelativistic theory). - 4th edition. - M .: Nauka , 1989 .-- 768 p. - (“ Theoretical Physics ”, Volume III). - ISBN 5-02-014421-5 . - mathematical additions
  • Kazuhiko Aomoto, Michitake Kita. Theory of Hypergeometric Functions / Transl. by Kenji Iohara. - Springer, 2011 .-- Vol. 305. - 317 p. - (Springer Monographs in Mathematics Series). - ISBN 9784431539124 .
  • Scott JF The mathematical work of John Wallis, DD, FRS, (1616-1703). - American Mathematical Soc., 1981. - 240 p. - (Chelsea Publishing Series). - ISBN 9780828403146 .
Source - https://ru.wikipedia.org/w/index.php?title= Hypergeometric_function&oldid = 102186252


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