HermiteHGeneral. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation

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1 HermiteHGeeral Notatios Traditioal ame Hermite fuctio Traditioal otatio H Mathematica StadardForm otatio HermiteH, Primary defiitio H F ; ; F ; 3 ; Specific values Specialied values For fixed H 0 For fixed Explicit iteger H H H 4

2 H H H H H H H H Symbolic iteger H k k k k ; H H k k k L k k k j j erf k j 0 k 3 j L j j j j 4 ; Geeral characteristics Domai ad aalyticity H is a aalytical fuctio of ad which is defied over. For fixed, it is a etire fuctio of. For fixed, it is a etire fuctio of. For iteger, H degeerates to a polyomial i H

3 3 Symmetries ad periodicities Parity H H ; Mirror symmetry H H Periodicity No periodicity Poles ad essetial sigularities With respect to For fixed ;, the fuctio H has oly oe sigular poit at. It is a essetial sigular poit ig H, ; For positive iteger, the fuctio H is polyomial ad has pole of order at ig H, ; With respect to For fixed, the fuctio H has oly oe sigular poit at. It is a essetial sigular poit ig H, Brach poits With respect to For fixed, the fuctio H does ot have brach poits H With respect to For fixed, the fuctio H does ot have brach poits H Brach cuts With respect to

4 4 For fixed iteger, the fuctio H does ot have brach cuts H With respect to For fixed, the fuctio H does ot have brach cuts H Series represetatios Geeralied power series Expasios at geeric poit 0 For the fuctio itself H H 0 H 0 0 H 0 0 ; H H 0 H 0 0 H 0 0 O 0 3 H k k k H k 0 0 k k k H k 0 F, ; k, k k k ; 0 0 F, ; k, 3 k ; 0 0 k H H 0 O 0 Expasios at 0 For the fuctio itself Geeral case H ; 0 30

5 5 H H O O 6 30 k k k k k k 3 k k H F ; ; F ; 3 ; H O O H F, ; F m, m k k k k m k k 3 k k m H m m m F, m ; m 3, m ; m 3 m F, m 3 ; m, m 5 ; m Summed form of the trucated series expasio. Special cases H k k k k ; H O ; Expasios at For the fuctio itself Special cases

6 H ; H O ; H k k k k k ; H k k k k ; H F 0, H O ; ; ; ; Asymptotic series expasios si H 4 csc sec ; H O arg O O ; 3 4 O arg O True

7 si H 4 csc sec k k k k k O k k k k O ; k k k k k O arg H k k k k k O k k k k O arg ; k k k k k O k k k k O True si H F 0, ; ; csc sec 4 F 0, ; ; ; H F 0, ; ; arg F 0, ; ; F 0, ; ; arg F 0, ; ; F 0, ; ; True ; H O 4 cos si O ; H O arg O O arg O O True ; H F 0, ; ; ; arg

8 H O ; arg Itegral represetatios O the real axis H t t cos t H 3 t t t ; t ; Re Itegral represetatios of egative iteger order Rodrigues-type formula H ; Limit represetatios H lim Λ L Λ Λ Λ Λ H lim Λ Λ C Λ Λ ; H lim a a,a P a a Geeratig fuctios H t tt ; Differetial equatios Ordiary liear differetial equatios ad wroskias w w w 0 ; w c H c H

9 W H, H w w w 0 ; w c H c F ; 3 ; F ; ; W H, F ; 3 ; F ; ; w w w 0 ; w c H c F ; ; W H, F ; ; w w w 0 ; w c H c F ; 3 ; W H, F ; 3 ; w g g g g w g w 0 ; w c H g c g H g W H g, g H g g g w g g h h g g w w c h H g c h g H g g g h g h W h H g, h g H g g h g h g h g h h h h w 0 ; w a r r r s w a r s r r s r s w 0 ; w c s H a r c s a r H a r W s H a r, s a r H a r a a r r s r

10 w a r logr logs w a logr logr logs r logs logr logs w 0 ; w c s H a r c s a r H a r W s H a r, s a r H a r a a r r s logr Trasformatios Trasformatios ad argumet simplificatios Argumet ivolvig basic arithmetic operatios H H ; arg H H si L H H F ; 3 ; H H ; H H si L H H F H H ; ; 3 ; Additio formulas k H k H k k k k H H k k

11 H k H k H k ; H cosα siα cos k taα k Α H k H k k cos k Α si k Α H k H k H cosα siα ; k k Multiple argumets H k k k H k ; Products, sums, ad powers of the direct fuctio Products of the direct fuctio mi,m k H km H H m m ; m k k m k mi,m H H m m k k H km k k ; m Idetities Recurrece idetities Cosecutive eighbors H H H H H H Distat eighbors H, H, H ; 0,,,,,

12 H, H, H ; 0,,,,, Fuctioal idetities Relatios betwee cotiguous fuctios Recurrece relatios H H H H H H Normalied recurrece relatio p, p, p, ; p, H Complex characteristics Real part j j j y j ReH x y j 0 H j x ; x y j Imagiary part j j j y j ImH x y H j x ; x y j j 0 Differetiatio Low-order differetiatio With respect to

13 H log H 4 Ψ 6 3 F 0 F ; ; ; ;, ;, 3 ;; ;, F 0 Ψ F 3 ; ;, ;,, 5 3 ;; ; ; 3 ; H log H 4 Ψ F ; ; Ψ F ; 3 ; Ψ F ; Ψk ; k k k k Ψ F ; 3 ; Ψk k k k 3 k H log H Ψ log6 Ψ Ψ F ; ; Ψ log6 Ψ Ψ F ; 3 ; log4 Ψ Ψ F ; ; k Ψk k k k Ψ Ψ F ; ; k k k k Ψ k Ψ Ψ k Ψ k 4 log4 Ψ Ψ F ; 3 ; Ψk k k k 3 k Ψ Ψ F ; 3 ; k k k 3 k Ψ k Ψ Ψ k Ψ k

14 H,0 F, ; 3, ; erfi Ψ H Ψ k 4k k k k k L k k k k k L k L k ; Brychkov Yu.A. (006) H,0 0 F, ; 3, ; erfi Brychkov Yu.A. (006) H,0 F, ; 3, ; erfc erfi Brychkov Yu.A. (006) H With respect to log4ψ H H H 4 H Backward shift operator: H H H True H H Symbolic differetiatio With respect to

15 5 m H m m m k log mk j 0 k j k j k j j 0 j k j j ; m With respect to m H m m m H m ; m m H m m m F, ; m, m ; m m F, ; m, 3 m ; ; m Fractioal itegro-differetiatio With respect to Α H Α Α Α F, ; Α, Α ; Α Α F, ; Α, 3 Α ; Itegratio Idefiite itegratio Ivolvig oly oe direct fuctio H H Ivolvig oe direct fuctio ad elemetary fuctios Ivolvig power fuctio Α H Α Α F, Α ;, Α ; Α F, Α ; 3, Α 3 ; H 3 3 F ; ; 4 3 F ; 5 ;

16 m H a m m F, m a a m F, m ; 3, m ; a H F ; ; F ; 3 ; Ivolvig expoetial fuctio H F ; 3 ; F ; ; Ivolvig expoetial fuctio ad a power fuctio Α p H Α p H Α a H a Α p Α Α Α p k Α, p k 3 k k p k Α k, p k 3 k pk k Α p Α Α p Α k Α, p k k k p k k Α k, p k pk k Α Α F , Α ;, Α a Α ; a F H si F Ivolvig fuctios of the direct fuctio ad elemetary fuctios, Α ; 3, Α 3 ; a ; 3 ; F ; ; Ivolvig elemetary fuctios of the direct fuctio ad elemetary fuctios Ivolvig powers of the direct fuctio ad a power fuctio H 8 H H

17 7 Ivolvig powers of the direct fuctio, power ad expoetial fuctios H 4 H H Ivolvig direct fuctio ad Gamma-, Beta-, Erf-type fuctios Ivolvig probability itegral-type fuctios Ivolvig erf erfa H a a a H a erfa H a Ivolvig erfi erfi H H erfi H Defiite itegratio Ivolvig the direct fuctio t Α a t H tt a Α 0 Α F Α, ; ; a Α a Α a F Α, ; ; a a Α Α F, ; ; a ; Rea 0 ReΑ t H tt ; Orthogoality: t H m t H tt,m ; m t H l t H m t H tt lm l m l m l m l m ; l m l m l m m l l m

18 t H l t H m t H tt 0 ; l m l m l m m l l m Summatio Fiite summatio H k x H k y H x H y H x H y ; k k x y k k k k H k x H k y x y ; cos k H k x H k y Rex y ; si k H k x H k y Imx y ; Ifiite summatio H w ww H w w cos 0 w H w w si 0 w w c H w w c F c; ; w w c H w 0 w c F c ; 3 ; w w H w w w 4 w 3 exp 4 w 4 w

19 H x y t x ΑΑ W t y ; Α t y y Α y H kj w kj j 0 k j j l l 4 l exp j w j w k l j ; k j H H w 0 4 w exp w w 4 w ; w H x H y 0 x y x y ; x y m H H m m H m ; m Operatios Limit operatio lim 4 H si ; lim H cos ; Orthogoality, completeess, ad Fourier expasios The set of fuctios H x, 0,,, forms a complete, orthogoal (with weight iterval, x ) system o the 0 x H x y H y x y m m t H m t t H t t,m

20 0 Ay sufficietly smooth fuctio f x ca be expaded i the system H x 0,, as a geeralied Fourier series, with its sum covergig to f x almost everywhere f x c Ψ x ; c Ψ t f tt Ψ x 0 x H x ; x Represetatios through more geeral fuctios Through hypergeometric fuctios Ivolvig F H F ; ; F ; 3 ; Ivolvig F H F ; ; F ; 3 ; Ivolvig p F q H F 0, Ivolvig hypergeometric U ; ; ; H U,, ; Re H U, 3, ; Re H F ; ; U,, H U, 3, F ; ; H F ; 3 ; U,,

21 H F ; 3 ; U, 3, Through Meijer G Classical cases for the direct fuctio itself H G,, 0, ; H 3 G, 3,4, 8, 5 8 0,, 8, H G,, 0, ; arg H H 3 G,,3, 4 0,, H H 3 G,,3, 4, 0, 4 Classical cases ivolvig exp H G,0, 0, ; arg H G,0, 0, H G,,3, 0,, H H cos G,, 0,

22 H H si G,,, 0 Classical cases ivolvig exp ad cosh cosh H G,0, 0, G,, 4 0, ; arg cosh, H G,,3 0,, 3,, 5 G, 3, ,,, ; arg cosh H H cos G,, 0, 3, G,,3 4 0,, 4 ; arg cosh H H 3, G,,3 4, 0, 4 G,, si, 0 ; arg cosh H G,0, 0, G,, 4 0, cosh H G,,3, 0,, 3 G, 3,4, 8, 5 8 0,, 8, cosh H H cos G,, 0, 3 G,,3, 4 0,, cosh H H 3 G,,3, 4, 0, 4 G,, si, 0 Classical cases ivolvig exp ad sih sih H 4 G,, 0, G,0, 0, ; arg

23 sih H 3,, 5 G, 3, ,,, 5 8 8, G,,3 0,, ; arg sih H H 3, G,,3 4 0,, 4 cos G,, 0, ; arg sih H H 3, G,,3 4, 0, 4 si G,, ;, 0 arg sih H G,, 4 0, G,0, 0, sih H 3 G, 3,4, 8, 5 8 0,, 8, 5 8 G,,3, 0,, sih H H 3 G,,3, 4 0,, 4 cos G,, 0, sih H H 3 G,,3, 4, 0, 4 si G,,, 0 Classical cases for products of H H 4 4 H G 4,,4 4, 0, 4,, 3 4 Classical cases ivolvig Exp ad products of H 4 H H 4 3 4,0 G,4 4, 0, 4,, 3 4 Classical cases ivolvig Exp ad parabolic cylider D D H 3 G 4,,4 4 4, 0, 4,, 3 4 ; arg 0

24 D H 4,0 G 4,4 4, 0, 4,, 3 4 ; 4 arg 4 Geeralied cases for the direct fuctio itself H G,,, 0, ; H G,,, 3, ; H G, 3,4,, 8, 5 8 0,, 8, H G,0,,, ; H lim m m G,,, m3, ; H H 3 G,,3,, 4 0,, H H 3 G,,3,, 4, 0, 4 Geeralied cases ivolvig exp H G,0,, 0, H G,,3,, 0,, H H cos G,,, 0,

25 H H si G,,,, 0 Geeralied cases ivolvig exp ad cosh cosh H G,0,, 0, G,,, 4 0, cosh H G,,3,, 0,, 3 G, 3,4,, 8, 5 8 0,, 8, cosh H H cos G,,, 0, 3 G,,3,, 4 0,, cosh H H 3 G,,3,, 4, 0, 4 si G,,,, 0 Geeralied cases ivolvig exp ad sih sih H 4 G,,, 0, G,0,, 0, sih H 3 G, 3,4,, 8, 5 8 0,, 8, 5 8 G,,3,, 0,, sih H H 3 G,,3,, 4 0,, 4 cos G,,, 0, sih H H 3 G,,3,, 4, 0, 4 si G,,,, 0 Geeralied cases for products of H H 4 H 4 4, G,4, 4, 0, 4,, 3 4 Geeralied cases ivolvig Exp ad products of H

26 H H 4,0 G,4,, 0, 4,, H H 4,0 G,4, 4, 0, 4,, 3 4 Geeralied cases ivolvig Exp ad parabolic cylider D D H 3 G 4,,4 4, 4, 0, 4,, D H 4,0 G,4, 4, 0, 4,, 3 4 Through other fuctios Ivolvig some hypergeometric-type fuctios H cos L si L H L ; H L ; H lim Λ L Λ Λ Λ Λ H lim Λ Λ C Λ Λ ; H lim a a,a P a a Represetatios through equivalet fuctios With related fuctios

27 H cos L si H L ; L H L ; H D Zeros x j ; H x k 0 x k j x k k j Theorems Expasios i geeralied Fourier series f x c k Ψ k x ; c k f tψ k tt, Ψ k x k 4 k x H k x, k. Fourier trasform eigefuctios Hermite polyomials together with their weightig fuctio are eigefuctios of the Fourier ad iverse Fourier trasforms: t x x H x x t H t ; Zeros of Hermite polyomials For ay give iterval a, b, there exists some such that H x has a ero i this iterval. The umber of simple graphs The umber of simple graphs with o cycles ad vertices is H H. History

28 8 P. S. Laplace (80) Ch. Hermite (864) N. J. Soie (880)

29 9 Copyright This documet was dowloaded from fuctios.wolfram.com, a comprehesive olie compedium of formulas ivolvig the special fuctios of mathematics. For a key to the otatios used here, see Please cite this documet by referrig to the fuctios.wolfram.com page from which it was dowloaded, for example: To refer to a particular formula, cite fuctios.wolfram.com followed by the citatio umber. e.g.: This documet is curretly i a prelimiary form. If you have commets or suggestios, please commets@fuctios.wolfram.com , Wolfram Research, Ic.

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