U matematici Mnogochleni Bernulli mnogochleni nazvani na chest Yakoba Bernulli sho vinikayut pri vivchenni bagatoh specialnih funkcij zokrema z funkciyi Rimana i z funkciyi Gurvica takozh ye okremim vipadkom poslidovnosti Appelya Na vidminu vid ortogonalnih mnogochleniv mnogochleni Bernulli viznachni tim sho chislo koreniv v intervali 0 1 displaystyle 0 1 ne zbilshuyetsya iz zrostannyam stepenya mnogochlena Pri neobmezhenomu zbilshenni stepenya mnogochleni Bernulli nablizhayutsya do trigonometrichnih funkcij Podibnij nabir mnogochleniv zasnovanij na tvirnij funkciyi nazivayut simejstvom mnogochleniv Ejlera Zmist 1 Viznachennya 1 1 Yavna formula 1 2 Generatrisa 1 3 Predstavlennya diferencialnim operatorom 1 4 Viznachennya za dopomogoyu integralnogo operatora 1 5 Yavnij viglyad dlya najmenshih stepeniv 2 Vlastivosti 2 1 Znachennya v nuli 2 2 Diferenciyuvannya i integruvannya 2 3 Mnozhennya argumentu 2 4 Suma argumentu 2 5 Simetriya 3 Ryad Fur ye 4 Obertannya 5 Posilannya 6 LiteraturaViznachennya red Mnogochleni Bernulli B n x displaystyle B n x nbsp mozhna viznachiti riznimi sposobami Vibir viznachennya zalezhit vid zruchnosti v tomu abo inshomu vipadku Yavna formula red B n x k 0 n C n k B n k x k displaystyle B n x sum k 0 n C n k B n k x k nbsp de C n k displaystyle C n k nbsp binomialni koeficiyenti B k displaystyle B k nbsp chisla Bernulli Abo B n x m 0 n 1 m 1 k 0 m 1 k C m k x k n displaystyle B n x sum m 0 n frac 1 m 1 sum k 0 m 1 k C m k x k n nbsp Generatrisa red Generatrisa dlya mnogochleniv Bernulli rivna t e x t e t 1 n 0 B n x t n n displaystyle frac te xt e t 1 sum n 0 infty B n x frac t n n nbsp Generatrisa dlya mnogochleniv Ejlera rivna 2 e x t e t 1 n 0 E n x t n n displaystyle frac 2e xt e t 1 sum n 0 infty E n x frac t n n nbsp Predstavlennya diferencialnim operatorom red B n x D e D 1 x n displaystyle B n x D over e D 1 x n nbsp de displaystyle operator formalnogo diferenciyuvannya Viznachennya za dopomogoyu integralnogo operatora red Mnogochleni Bernulli ye yedinimi mnogochlenami sho zadovolnyayut rivnyannya x x 1 B n u d u x n displaystyle int x x 1 B n u du x n nbsp Integralnij operator T f x x x 1 f u d u displaystyle Tf x int x x 1 f u du nbsp dlya mnogochleniv f prijmaye ti zh znachennya sho j T f x e D 1 D f x n 0 D n n 1 f x f x f x 2 f x 6 f x 24 displaystyle begin aligned Tf x e D 1 over D f x amp sum n 0 infty D n over n 1 f x amp f x f x over 2 f x over 6 f x over 24 cdots end aligned nbsp Yavnij viglyad dlya najmenshih stepeniv red Mnogochlenami Bernulli dlya najmenshih stepeniv ye B 0 x 1 displaystyle B 0 x 1 nbsp B 1 x x 1 2 displaystyle B 1 x x frac 1 2 nbsp B 2 x x 2 x 1 6 displaystyle B 2 x x 2 x frac 1 6 nbsp B 3 x x 3 3 2 x 2 1 2 x displaystyle B 3 x x 3 frac 3 2 x 2 frac 1 2 x nbsp B 4 x x 4 2 x 3 x 2 1 30 displaystyle B 4 x x 4 2x 3 x 2 frac 1 30 nbsp B 5 x x 5 5 2 x 4 5 3 x 3 1 6 x displaystyle B 5 x x 5 frac 5 2 x 4 frac 5 3 x 3 frac 1 6 x nbsp B 6 x x 6 3 x 5 5 2 x 4 1 2 x 2 1 42 displaystyle B 6 x x 6 3x 5 frac 5 2 x 4 frac 1 2 x 2 frac 1 42 nbsp Vlastivosti red Znachennya v nuli red Znachennya mnogochleniv Bernulli pri x 0 displaystyle x 0 nbsp rivni vidpovidnim chislam Bernulli B n 0 B n displaystyle B n 0 B n nbsp Diferenciyuvannya i integruvannya red B n x n B n 1 x displaystyle B n x nB n 1 x nbsp Neviznacheni integrali a x B n t d t B n 1 x B n 1 a n 1 displaystyle int a x B n t dt frac B n 1 x B n 1 a n 1 nbsp Viznacheni integrali 0 1 B n t B m t d t 1 n 1 m n m n B n m m n 1 displaystyle int 0 1 B n t B m t dt 1 n 1 frac m n m n B n m quad m n geq 1 nbsp Mnozhennya argumentu red B n m x m n 1 s 0 m 1 B n x s m displaystyle B n mx m n 1 sum s 0 m 1 B n left x frac s m right nbsp Suma argumentu red B n x y k 0 n n k B k x y n k displaystyle B n x y sum k 0 n n choose k B k x y n k nbsp Simetriya red B n 1 x 1 n B n x displaystyle B n 1 x 1 n B n x nbsp 1 n B n x B n x n x n 1 displaystyle 1 n B n x B n x nx n 1 nbsp Ryad Fur ye red Ryadi Fur ye dlya mnogochleniv Bernulli ye takozh ryadami Dirihle B n x n 2 p i n k 0 e 2 p i k x k n 2 n k 1 cos 2 k p x n p 2 2 k p n displaystyle B n x frac n 2 pi i n sum k not 0 frac e 2 pi ikx k n 2n sum k 1 frac cos left 2k pi x frac n pi 2 right 2k pi n nbsp Cej rozklad spravedlivij koli 0 x 1 dlya n 2 i u vipadku 0 lt x lt 1 dlya n 1 Obertannya red x n 1 n 1 k 0 n n 1 k B k x displaystyle x n frac 1 n 1 sum k 0 n n 1 choose k B k x nbsp Posilannya red Kurtulan Ali Burak Bernoulli Polynomials and Their Applications nedostupne posilannya z lipnya 2019 Literatura red Grigorij Mihajlovich Fihtengolc Kurs diferencialnogo ta integralnogo chislennya 2023 1300 s ukr Milton Abramowitz and Irene A Stegun eds Handbook of Mathematical Functions with Formulas Graphs and Mathematical Tables 1972 Dover New York Apostol Tom M 1976 Introduction to analytic number theory Undergraduate Texts in Mathematics New York Heidelberg Springer Verlag ISBN 978 0 387 90163 3 Hugh L Montgomery Robert C Vaughan 2007 Multiplicative number theory I Classical theory Cambridge tracts in advanced mathematics 97 Cambridge Cambridge Univ Press pp 495 519 ISBN 0 521 84903 9 nbsp Ce nezavershena stattya z matematiki Vi mozhete dopomogti proyektu vipravivshi abo dopisavshi yiyi Otrimano z https uk wikipedia org w index php title Mnogochleni Bernulli amp oldid 40597800
