Some Generalizations of Vandermonde's Convolution

Gould, H. W.

doi:10.1080/00029890.1956.11988763

Cited by 108 publications

(59 citation statements)

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“…This is due to Jensen [17] when h = 1, and if h = 0 it reduces to (3.4). Some good references for Abel-Rothe type identities are [10,11,39]. This surprising formula has a number of uses in mathematics, among them the derivation of Abel-Rothe type identities.…”

Section: Abel-rothe Type Identitiesmentioning

confidence: 94%

The Pfaff/Cauchy derivative identities and Hurwitz type extensions

Johnson

2007

Ramanujan J

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More than 200 years ago, Pfaff found two generalizations of Leibniz's rule for the nth derivative of a product of two functions. Thirty years later Cauchy found two similar identities, one equivalent to one of Pfaff's and the other new. We give simple proofs of these little-known identities and some further history. We also give applications to Abel-Rothe type binomial identities, Lagrange's series, and Laguerre and Jacobi polynomials. Most importantly, we give extensions that are related to the Pfaff/Cauchy theorems as Hurwitz's generalized binomial theorems are to the Abel-Rothe identities. We apply these extensions to Laguerre and Jacobi polynomials as well.

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Section: Abel-rothe Type Identitiesmentioning

confidence: 94%

The Pfaff/Cauchy derivative identities and Hurwitz type extensions

Johnson

2007

Ramanujan J

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“…Due to this reason, extensions using multinomial coefficients may not be directly relevant to binary-valued 0-1 vector sets. Refined identity equations of combinatorics are required [14,15].…”

Section: Trinomial Coefficientsmentioning

confidence: 99%

“…Proof For a fixed p, 0 < p < n, all possible { f (q, p, n)} are collected to form the following combinatorial identities: [14,15,21],…”

Section: Combinatorial Projection On Variant Clustersmentioning

confidence: 99%

Symmetric Clusters in Hierarchy with Cryptographic Properties

Zheng

2018

Variant Construction From Theoretical Foundation to Applications

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Symmetric Boolean functions play a key role in stream ciphers. Symmetric constructions provide core components in cryptographic applications. In this chapter, four meta symmetric clustering schemes (combination, crossing, variant and rotation) are organized in a hierarchy for n variables of 0-1 vectors in measuring phase spaces. Local counting properties in a cluster and global counting properties in a given level are formulated. From selected symmetric clusters, a number of various symmetric Boolean functions are formulated. Counting properties on symmetric clusters, vectors in selected clusters and special symmetric Boolean functions are listed. Four sets of symmetric Boolean functions are compared. Properties of symmetric clusters and Boolean functions are discussed. Main results are expressed in theorems and tables. Among four meta schemes, the variant scheme presents novel properties approximately with O n 2 /4 clusters on a 2D phase space different from other schemes: combinatorial O (n), crossing O (n/2) and rotation O (2 n /n) on 1D measuring phase spaces, respectively. The variant pseudorandom number generator is a similar approach on RC4 and HC128 stream ciphers using word-oriented 0-1 vectors. Further advanced researches and explorations on relevant optimal configurations are required. Keywords Symmetric construction • Meta symmetric Cluster • hierarchy Boolean function • Four meta schemes • Phase space

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“…(13) ist ein Additionstheorem fiir den aus Binomialkoeffizienten aufgebauten Ausdruck Fur den Sonderfall ?n = I wurde das Theorem von GOULD [3] ohne Verweiiduiig cler LA4GRASGESCheli Reilio bewiesen. Ein dritter Beweis dieses Spezialfalles durcli Induktion iiber die Parameter a e und b ist dein Verfasser der vorliegenden Arbeit ebenfalls bekannt.…”

Section: Einleitnngunclassified

Über eine Anwendung der LAGRANGESchen Reihe

Tinhofer

1973

Mathematische Nachrichten

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Some Generalizations of Vandermonde's Convolution

Cited by 108 publications

References 0 publications

The Pfaff/Cauchy derivative identities and Hurwitz type extensions

The Pfaff/Cauchy derivative identities and Hurwitz type extensions

Symmetric Clusters in Hierarchy with Cryptographic Properties

Über eine Anwendung der LAGRANGESchen Reihe

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