Simvol Pohgammera poznachennya dlya specialnoyi funkciyi yaka zadayetsya dobutkom x n k 1 n x k 1 x x 1 x 2 x n 1 displaystyle x n prod k 1 n x k 1 x x 1 x 2 dots x n 1 de n displaystyle n nevid yemne cile chislo yakij she nazivayut zrostayuchim faktorialom Vikoristovuyetsya napriklad pri oznacheni gipergeometrichnoyi funkciyi V kombinatorici simvolom x n displaystyle x n poznachayut spadnij faktorial x n k 1 n x k 1 x x 1 x 2 x n 1 displaystyle x n prod k 1 n x k 1 x x 1 x 2 dots x n 1 a zrostayuchij faktorial simvolom x n displaystyle x n Nazva dana v chest nimeckogo matematika Leo Pohgammera Leo August Pochhammer Yaksho ne obumovleno okremo to nadali pid simvolom x n displaystyle x n rozumitimemo zrostayuchij faktorial Zmist 1 Prikladi 2 Vlastivosti 3 Zrostayuchij ta spadnij faktoriali 4 Uzagalnennya 5 PosilannyaPrikladi red Pershi dekilka znachen dlya nevid yemnih cilih n displaystyle n nbsp x 0 1 displaystyle x 0 1 nbsp x 1 x displaystyle x 1 x nbsp x 2 x 2 x displaystyle x 2 x 2 x nbsp x 3 x 3 3 x 2 2 x displaystyle x 3 x 3 3x 2 2x nbsp x 4 x 4 6 x 3 11 x 2 6 x displaystyle x 4 x 4 6x 3 11x 2 6x nbsp Chastkovi vipadki 1 n n 1 2 n 2 n 1 2 n displaystyle 1 n n qquad left frac 1 2 right n frac 2n 1 2 n nbsp Vlastivosti red Dlya simvoliv Pohgammera vikonuyetsya vidnoshennya x n 1 n x n 1 n displaystyle x n 1 n cdot x n 1 n nbsp Simvol Pohgammera mozhna viraziti cherez gamma funkciyu x n G x n G x displaystyle x n frac Gamma x n Gamma x nbsp ta cherez binomialnij koeficiyent x n n x n 1 n displaystyle frac x n n x n 1 choose n nbsp Simvol Pohgammera pov yazanij z chislami Stirlinga pershogo rodu s n k displaystyle s n k nbsp x n k 0 n 1 n k s n k x k displaystyle x n sum k 0 n 1 n k s n k cdot x k nbsp Spivvidnoshennya mizh simvolami Pohgammera dlya parnogo to neparnogo indeksu x 2 n 2 2 n x 2 n x 1 2 n x 2 n 1 2 2 n 1 x 2 n 1 x 1 2 n displaystyle x 2n 2 2n left frac x 2 right n left frac x 1 2 right n qquad x 2n 1 2 2n 1 left frac x 2 right n 1 left frac x 1 2 right n nbsp Vidnoshennya dvoh simvoliv Pohgammera x n x m x m n m n m 1 x n m n n m displaystyle frac x n x m left begin array ll x m n m amp n geqslant m displaystyle frac 1 x n m n amp n leqslant m end array right nbsp Pohidna simvolu Pohgammera d d x x n x n ps 0 n x ps 0 x displaystyle frac d dx x n x n cdot left psi 0 n x psi 0 x right nbsp de ps 0 x displaystyle psi 0 x nbsp digamma funkciya Zrostayuchij ta spadnij faktoriali red Tut budemo vikoristovuvati nastupni poznachennya prijnyati v kombinatorici Zrostayuchij faktorialx n x n x x 1 x 2 x n 1 displaystyle x n x overline n x x 1 x 2 cdots x n 1 nbsp Spadnij faktorial x n x n x x 1 x 2 x n 1 displaystyle x n x underline n x x 1 x 2 cdots x n 1 nbsp Spadnij faktorial chiselno dorivnyuye kilkosti rozmishen bez povtoren z x displaystyle x nbsp po n displaystyle n nbsp abo sho te same kilkosti usih in yektivnih funkcij z mnozhini potuzhnosti n displaystyle n nbsp v mnozhinu potuzhnosti x displaystyle x nbsp Zrostayuchij ta spadnij faktoriali pov yazani spivvidnoshennyami x n x n 1 n x n 1 n x n displaystyle x n x n 1 n quad x n 1 n x n nbsp Spadnij faktorial takozh mozhna viraziti cherez gamma funkciyu x n G x 1 G x n 1 displaystyle x n frac Gamma x 1 Gamma x n 1 nbsp ta cherez binomialnij koeficiyent x n n x n displaystyle frac x n n x choose n nbsp Za dopomogoyu spadnogo faktoriala mozhna kompaktno viraziti pohidnu n displaystyle n nbsp ogo poryadku vid stepenevoyi funkciyi d n d x n x a a n x a n displaystyle frac d n dx n x a a n x a n nbsp Formula dlya dobutku spadnih faktorialiv x m x n k 0 m m k n k k x m n k displaystyle x m x n sum k 0 m m choose k n choose k k x m n k nbsp Tvirna funkciya spadnogo faktorialu n 0 x n t n n 1 t x displaystyle sum n 0 infty x n frac t n n 1 t x nbsp Uzagalnennya red Simvol Pohgammera mozhna uzagalniti tak x n k x x k x 2 k x n 1 k displaystyle x n k x x k x 2k cdots x n 1 k nbsp i nazivayetsya k simvolom Pohgammera Simvol Pohgamera mozhna takozh uzagalniti na vipadok dovilnoyi funkciyi v takij formi f x k h f x f x h f x 2 h f x k 1 h displaystyle f x k h f x cdot f x h cdot f x 2h cdots f x k 1 h nbsp U takomu zapisi zvichajnij simvol Pohgammera zapisuyetsya yak x k 1 displaystyle x k 1 nbsp Takozh u kombinatorici vikoristovuyetsya q analog simvolu Pohgammera abo q simvol Pohgammera ne plutati z k simvolom x q n k 0 n 1 1 x q k 1 x 1 x q 1 x q 2 1 x q n 1 displaystyle x q n prod k 0 n 1 1 xq k 1 x 1 xq 1 xq 2 cdots 1 xq n 1 nbsp Posilannya red Weisstein Eric W Pochhammer Symbol angl na sajti Wolfram MathWorld Weisstein Eric W Falling Factorial angl na sajti Wolfram MathWorld Otrimano z https uk wikipedia org w index php title Simvol Pohgammera amp oldid 40276000
