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Roman S. — The Umbral Calculus
Roman S. — The Umbral Calculus

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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: en

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

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
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 $xs_n(x)$      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 $p’_n(x)$      26
Associated sequence(s), expansion of $xp_n(x)$      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 $s’_n(x)$      24
Sheffef sequence, expansion of $xs_n(x)$      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
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