Advances in Combinatorics: Waterloo Workshop in Computer by Gert Almkvist (auth.), Ilias S. Kotsireas, Eugene V. Zima

By Gert Almkvist (auth.), Ilias S. Kotsireas, Eugene V. Zima (eds.)

This quantity, as Andrew M. Odlzyko writes within the foreword, “commemorates and celebrates the lifestyles and achievements of a rare person.” initially conceived as an eightieth birthday tribute to Herbert Wilf, the well known combinatorialist, the e-book has developed past the proceeds of the W80 tribute.

Professor Wilf used to be an award-winning instructor, who was once supportive of girls mathematicians, and who had an surprisingly excessive percentage of ladies between his PhD applicants. He was once Editor-in-chief of the yankee Mathematical per month and a founding father of either the magazine of Algorithms and of the digital magazine of Combinatorics. yet he used to be first a researcher, pushed through his wish to be aware of and clarify the internal workings of the mathematical world.

The e-book collects fine quality, refereed study contributions via a few of Professor Wilf’s colleagues, scholars, and collaborators. a few of the papers provided the following have been featured within the 3rd Waterloo Workshop on desktop Algebra (WWCA 2011, W80), held could 26-29, 2011 at Wilfrid Laurier collage, Waterloo, Canada. Others have been integrated as a result of their dating to his very important paintings in combinatorics. All are awarded as a tribute to Herb Wilf’s contributions to arithmetic and mathematical life.

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Additional resources for Advances in Combinatorics: Waterloo Workshop in Computer Algebra, W80, May 26-29, 2011

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1/k ˆ k 1 ! Á 0 mod p 3 C > > 2n nDk > > 162n > ; 2k 2k . 1/k k k n 18 G. 2n C 1/2 1 n 2n . 2 2n 2k 2k n . n; k/ D ! 2n 2k 1/2 4k 2k ! 4 ! 4 ! n/ D 968704n7 C 2683904n6 C 3013376n5 C 1758208n4 C568224n3 C 100200n2 C 8844n C 315: Glaisher’s Formulas for 1 2 and Some Generalizations 19 References 1. G. Bauer, Von den Coefficienten der Kugelfunctionen einer Variablen, J. Reine Angew. Math. 56 (1859), 101–129. formulae, Math.

Murty [11] in reference to p-adic irrationality. Q Y. n/ ¤ 0. n/ D 0 has at most two solutions. This has been achieved by different techniques by N. C. Alexander [1] and Junkyu An [5]. Our interest in the non-vanishing questions comes from the theory of summation in finite terms. The methods developed by R. Gosper show that the finite sum n X kŠ (8) kD1 does not admit a closed-form expression as a hypergeometric function of n. The identity n 1 X kD1 k a kŠ D a n 1 X X Q C 1/ . a/ C . n C 1/Š 1 (11) 26 T.

Z/ 2z nCz z ! 3z/ . 3z/ . 2 ˇ 1 3ˇ / ˇ 2 ˇˇ Ä jyj6xC1 Ä jyj13 ˇ z/3 ˇ Glaisher’s Formulas for 1 2 and Some Generalizations 15 Furthermore ˇ ˇ ˇ 1 3 ˇˇ ˇ . 1 C z C n/3 ˇ ˇ ˇ 1 1 Ä 15=2 for large n n3=2C6x n We have ! 4n C 1/ n 4n 43n . n; z/j Ä 4n 4 jyj13 1 43n 1 13 Ä jyj 3 2 n15=2 43n . c jyj// 3 5=2 2 for any positive c < 2 . z/ is constant. 0; z/ ! 3 2 1 1 t u when z ! n; / D 0 for n > 0. 2 2 ! 2k 2n nCk nCk ! with . 1/k n ! and we Remark 5. For n < k we must replace 2n 2k 2n 2k k n obtain the formulas (i) .

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