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Название: The Umbral Calculus

Автор: Roman S.

Аннотация:

Geared toward upper-level undergraduates and graduate students, this elementary introduction to classical umbral calculus requires only an acquaintance with the basic notions of algebra and a bit of applied mathematics (such as differential equations) to help put the theory in mathematical perspective. Subjects include Sheffer sequences and operators and their adjoints, with numerous examples of associated and other sequences. Related topics encompass the connection constants problem and duplication formulas, the Lagrange inversion formula, operational formulas, inverse relations, and binomial convolution.

Язык: Рубрика: Математика/Алгебра/Комбинаторика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1984

Количество страниц: 193

Добавлена в каталог: 08.03.2005

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 Abel function      11 72 85 Abel operator      14 Abel polynomials      72—75 Abel polynomials, binomial identity      73 Abel polynomials, expansion theorem      74 Abel polynomials, generating function      72 Abel polynomials, inverse under umbral composition      75 Abel polynomials, transfer formula      72 Actuarial polynomials      123—125 140 Actuarial polynomials, generating function      123 Actuarial polynomials, recurrence relation      124 Actuarial polynomials, Sheffer identity      123 Actuarial polynomials, umbral composition      125 Adjoint, of linear operator on P      34 Appell cross sequence      140 Appell identity      27 see Appell sequence      17 26—28 164 Appell sequence, Appell identity      27 Appell sequence, conjugate representation      27 Appell sequence, expansion of 27 Appell sequence, expansion theorem      26 Appell sequence, generating function      27 Appell sequence, multiplication theorem      27 Appell sequence, polynomial expansion theorem      26 Associated sequence(s)      17 25—26 48 50—51 164 Associated sequence(s), action of operator h(t) on      26 Associated sequence(s), binomial identity      26 Associated sequence(s), conjugate representation      25 Associated sequence(s), expansion of 26 Associated sequence(s), expansion of 26 Associated sequence(s), generating function      25 Associated sequence(s), operator characterization of      25 Associated sequence(s), polynomial expansion theorem      25 Associated sequence(s), recurrence formulas      48 Associated sequence(s), relationship between      51 Associated sequence(s), transfer formulas      50 Automorphisms of P*      33—36 38 Backward difference functional      63 Bell polynomials      82—86 Bell polynomials, connection with Abel function      85 Bell polynomials, connection with exponential polynomials      85 Bell polynomials, connection with idempotent numbers      85 Bell polynomials, connection with Laguerre polynomials      86 Bell polynomials, connection with Lah numbers      86 Bell polynomials, connection with Stirling numbers      85 Bernoulli numbers      12 94 98—100 Bernoulli numbers of second kind      114 Bernoulli numbers, connection with harmonic series      99—100 Bernoulli numbers, connection with Stirling numbers      99 Bernoulli polynomials      12 93—100 105 117—118 129 135 140 151 Bernoulli polynomials of second kind      113—119 Bernoulli polynomials of second kind, connection with Bernoulli polynomials      117—118 Bernoulli polynomials of second kind, expansion theorem      116 Bernoulli polynomials of second kind, generating function      116 Bernoulli polynomials of second kind, Gregory's formula      117 Bernoulli polynomials of second kind, polynomial expansion theorem      117 Bernoulli polynomials of second kind, Sheffer identity      116 Bernoulli polynomials of second kind, symmetry of      119 Bernoulli polynomials, Appell identity      94 Bernoulli polynomials, complementary argument theorem      100 Bernoulli polynomials, connection with Bernoulli polynomials of second kind      117 118 Bernoulli polynomials, connection with Euler polynomials      105 135 Bernoulli polynomials, connection with lower factorial polynomials      93 Bernoulli polynomials, connection with Stirling polynomials      129 Bernoulli polynomials, Euler —Maclautin expansion      97 Bernoulli polynomials, expansion theorem      96 Bernoulli polynomials, generating function      94 Bernoulli polynomials, multinomial theorem      97 Bernoulli polynomials, polynomial expansion theorem      93 Bernoulli polynomials, recurrence formula      95 Bernoulli polynomials, transfer formula      96 100 Bessel polynomials      78—82 Bessel polynomials, binomial identity      79 Bessel polynomials, generating function      79 Bessel polynomials, polynomial expansion theorem      80 Bessel polynomials, recurrence formula      80 Bessel polynomials, transfer formula      78—79 Binomial convolution      160—161 Binomial identity      26 see Binomial type, sequence of      26 Boole polynomials      127 Boole's summalion formula      104 Central difference series      68 see Chebyshev polynomials      162 166 173 Complementary argument theorem for Bernoulli polynomials      100 Complementary argument theorem for Euler polynomials      106 Conjugate representation      see also specific polynomial sequence Conjugate representation of Appell sequences      27 Conjugate representation of associated sequences      25 Conjugate representation of Sheffer sequences      20 Connection constants problem      131—138 Cross sequences      140 Delta functional      10 Delta operator      13 Delta series      4 Delta series, genetic      82 Derivations on P*      34 Derivations on P*, chain rule for      46 Dobinski's formula      66 Duplication formulas      132 137—138 Duplication formulas for Hermite polynomials      137 Duplication formulas for Laguerre polynomials      137 Duplication formulas for Mittag — Leffler polynomials      138 Euler numbers      102 Euler polynomials      12 100—106 135—136 161 Euler polynomials, Appell identity      102 Euler polynomials, Boole's summation formula      104 Euler polynomials, complementary argument theorem      106 Euler polynomials, connection with Bernoulli polynomials      105 135—136 Euler polynomials, expansion theorem      104 Euler polynomials, generating function      102 Euler polynomials, multiplication theorem      105 Euler polynomials, Newlon’s expansion of      101 Euler polynomials, recurrence formula      103 Evaluation functional      7 11 163 Expansion theorem for Appell sequences      26 Expansion theorem for associated sequences      25 Expansion theorem for Sheffer sequences      18 164 Exponential polynomials      63—67 85 Exponential polynomials, binomial identity      64 Exponential polynomials, connection with actuarial polynomials      123 Exponential polynomials, connection with Laguerre polynomials      134 Exponential polynomials, connection with Poisson — Charlief polynomials      122 Exponential polynomials, Dobinski's formula      66 Exponential polynomials, expansion theorem      65 Exponential polynomials, generating function      64 Exponential polynomials, recurrence formula      64 f      3 7 12 Falling factorial polynomials      see Lower factorial polynomials Fibonacci numbers      149 Formal power series      3—4 Forward difference functional      11 56 Forward difference operator      14 Gaussian coeffcient      see q-Umbral calculus Gegenbauer polynomials      162 166 173 Generalized Appell type      19 164 Generating function      see also specific polynomial sequence Generating function for Appell sequences      27 Generating function for associated sequences      25 Generating function for Sheffer sequences      18—19 Generic associated sequence      82—84 see Generic associated sequence, binomial identity      83 Generic associated sequence, conjugate representation      82 Generic associated sequence, generating function      83 Generic associated sequence, recurrence formula      83—84 Gould polynomials      67—72 149 Gould polynomials, binomial identity      69 Gould polynomials, expansion theorem      70 Gould polynomials, generating function      68 Gould polynomials, recurrence formula      69 Gould polynomials, Stirling's formula      70 Gould polynomials, Vandermonde's convolution formula      69 Gregory's formula      59 117 Heine's theorem      see q-Umbral calculus Hermite polynomials      87—93 135 137 140 150 158 Hermite polynomials, Appell identity      88 Hermite polynomials, classical Rodrigues formula      89 Hermite polynomials, duplication formula      137 Hermite polynomials, generating function      88 Hermite polynomials, multiplication theorem      93 Hermite polynomials, polynomial expansion theorem      89—90 Hermite polynomials, q-Hermite polynomials      see q-Umbral calculus Hermite polynomials, recurrence formula      89 Hermite polynomials, recurrence relation for      158 Hermite polynomials, squared Hermite polynomials      128 Hermite polynomials, urnbrai composition of      92 Hermite polynomials, Weierstrass operator      88 Idempotent numbers      85 Integration by parts      118 Inverse relations      147—156 Invertible functional      10 Invertible operator      13 Krawtchouk polynomials      126 Lagrange inversion formula      138—140 Laguerre polynomials      11 108—113 120 133—134 137 140 151 158—159 Laguerre polynomials, classical Rodrigues formula      109 Laguerre polynomials, connection with exponential polynomials      134 Laguerre polynomials, connection with Poisson — Charlier polynomials      120 Laguerre polynomials, duplication formula      137 Laguerre polynomials, generating function      110 Laguerre polynomials, Lab numbers      109 Laguerre polynomials, polynomial expansion theorem      112 Laguerre polynomials, recurrence formula      110 Laguerre polynomials, recurrence relation for      158—159 Laguerre polynomials, second order differential equation for      110—111 Laguerre polynomials, Sheffer identity      110 Laguerre polynomials, transfer formula      109 Laguerre polynomials, umbral composition of      112—113 Lah numbers      86 109 Linear operator, continuous      33 Lower factorial polynomials      56—63 Lower factorial polynomials, binomial identity      57 Lower factorial polynomials, conjugate representation      61—62 Lower factorial polynomials, expansion theorem      58 Lower factorial polynomials, generating function      57 Lower factorial polynomials, Gregory's formula      59 Lower factorial polynomials, Newton's forward difference interpolation formula      58 Lower factorial polynomials, polynomial expansion theorem      59 Lower factorial polynomials, recurrence formula      56 58 Lower factorial polynomials, Vandermonde convolution formula      57 Mahler polynomials      129—130 Meixner polynomials of first kind      125—126 Meixner polynomials of second kind      126 Mittag — Leffler polynomials      75—77 138 Mittag — Leffler polynomials, binomial identity      76 Mittag — Leffler polynomials, duplication formula      138 Mittag — Leffler polynomials, expansion theorem      77 Mittag — Leffler polynomials, generating function      76 Mittag — Leffler polynomials, recurrence formula      77 Mott polynomials      130 Multiplication theorem      27 93 97—98 105 Multiplication theorem for Bernoulli polynomials      97—98 Multiplication theorem for Euler polynomials      105 Multiplication theorem for Hermite polynomials      93 Narumi polynomials      127 Newton's forward difference polynomials      58 Operational formulas      144—147 Orthogonal polynomials, three term recurrence relation for      156 159—160 P      6 Peters polynomials      128 Pidduck polynomials      126—127 Pincherele derivative      49 Poisson dislribution      119 Poisson — Chadier polynomials      119—123 136 140 Poisson — Chadier polynomials, connection with Laguerre polynomials      120 Poisson — Chadier polynomials, generating function      120 Poisson — Chadier polynomials, recurrence formula      121 Poisson — Chadier polynomials, Sheffer identity      121 Poisson — Chadier polynomials, umbral composition of      122 Polynomial expansion theorem      see also specific polynomial sequence Polynomial expansion theorem for Appell sequences      26 Polynomial expansion theorem for associated sequences      25 Polynomial expansion theorem for Sheffer sequences      18 164 Poweroid      19 q-binomial coefficient      see q-Umbral calculus q-derivative      see q-Umbral calculus q-Hermite polynomials      see q-Umbral calculus q-Leibniz formula      see q-Umbral calculus q-Umbral calculus      174- 182 q-Umbral calculus, Gaussian coefficients      175 q-Umbral calculus, Heine's theorem      179 q-Umbral calculus, q-binomial coefficienls      175—177 q-Umbral calculus, q-derivative      175 q-Umbral calculus, q-Hermite polynomials      180—181 q-Umbral calculus, q-Hermite polynomials, generating function      180 q-Umbral calculus, q-Hermite polynomials, inverse under urnbrai composition      180 q-Umbral calculus, q-Hermite polynomials, polynomial expansion theorem      18 i q-Umbral calculus, q-Leibniz formula      176 Recurrence formulas for associated sequences      48 Recurrence formulas for Sheffef sequences      50 166 Rising factorial polynomials      63 Sheffef identily      21 see Sheffef sequence      17—24 164—165 Sheffef sequence, action of operator of form h(t) on      21 Sheffef sequence, characterization by Sheffer identity      21 165 Sheffef sequence, conjugate representation of      19—20 164 Sheffef sequence, expansion of 24 Sheffef sequence, expansion of 24 Sheffef sequence, expansion theorem      18 164 Sheffef sequence, generating function      18—19 164 Sheffef sequence, operalor characterization of      20 165 Sheffef sequence, performance with respect to multiplication in urnbrai algebra      23 Sheffef sequence, polynomial expansion theory      18 164 Sheffef sequence, recurrence formulas      50 Sheffef shift      49- 50 Sheffer operators      42—44 Sheffer operators, adjoint of      42 Sheffer operators, group of      43 Signless Stirling numbers of first kind      63 Squared Hermite polynomials      128 Steffensen sequences      140—143 Stirling numbers of first kind      57—59 61—62 66—67 85 99 115 129 134 Stirling numbers of first kind, connection with Stirling polynomials      129 Stirling numbers of first kind, signless      63 Stirling numbers of second kind      59—60 62—64 66—67 85 99 129 144 Stirling numbers of second kind, connection with Stirling polynomials      129 Stirling polynomials      128—129 Stirling polynomials, connection with Bernoulli polynomials      129 Stirling polynomials, connection with Stirling numbers      129 Stirling polynomials, generating function      128 Stirling's formula      70 Transfer formulas      50 166 Transfer operator      165 Translation operator      14 Umbral algebra      7 Umbral composition      41 Umbral operator      37—42 see Umbral operator, adjoint of      38 Umbral operator, group of      40 Umbral shift(s)      45—49 165 Umbral shift(s), adjoint of      46 Umbral shift(s), formula for      48 Umbral shift(s), relationship between      47 Vandermonde convolution formula      57 69 Реклама     © Электронная библиотека попечительского совета мехмата МГУ, 2004-2019 | | О проекте