bib.tex

Concrete_Mathematics_2nd_Ed_TeX_Source_Code

B.\,F. "Logan", ``Polynomials related to the Stirling numbers,'' AT\&T Bell
Laboratories internal technical memorandum, August 10, 1987.

\bib|long-hoggatt|%
Calvin T. "Long" and Verner E. "Hoggatt", Jr., ``Sets of binomial coefficients
with equal products,'' {\sl Fibonacci Quarterly\/ \bf12} (1974), 71--79.

\bib|lou-yao|%
Shituo "Lou" and Qi "Yao", ``A Chebychev's % sic
type of prime number theorem in a short interval-II,''
{\sl Hardy-Ramanujan Journal\/ \bf15} (1992), 1--33.
%``On upper bound of difference between
%consecutive primes,'' {\sl Kexue Tongbao [=K'o Hs\"ueh T'ung Pao]\/ \bf30},8
%(1985), Foreign language edition, 1127--1128.

\bib|loyd-cyclopedia|%
Sam "Loyd", {\sl Cyclopedia of Puzzles}. \
% Green cloth cover says ``Sam Loyd's Cyclopedia of 5,000 Puzzles, Tricks,
% and Conundrums, with Answers'', but the title page reads as cited.
Franklin Bigelow Corporation, Morningside Press, New York, 1914.

\bib|lucas-gcd|%
E. "Lucas", ``Sur les rapports qui existent entre la th\'eorie des nombres
et le Calcul int\'egral,''
 {\sl Comptes Rendus hebdomadaires des s\'eances de l'Acad\'emie des
 Sciences\/} (Paris) {\bf82} (1876), 1303--1305.

\bib|lucas-bc-mod|%
\'Edouard "Lucas", ``Sur les congruences des nombres eul\'eriens et des
coefficients diff\'erentiels des fonctions trigonom\'etriques, suivant un
module premier,''
{\sl Bulletin de la Soci\'et\'e math\'ematique de
France\/ \bf6} (1878), \hbox{49--54}.

\bib|lucas-theorie|%
Edouard "Lucas", {\sl Th\'eorie des Nombres}, volume~1. \ % sic, no accent
Gauthier-Villars, Paris, 1891.

\bib|lucas-recr|%
\'Edouard "Lucas", {\sl R\'ecr\'eations math\'ematiques}, four volumes. \
Gauthier-Villars, Paris, 1891--1894. Reprinted by Albert Blanchard, Paris, 1960.
\ (The Tower of Hanoi is discussed in volume~3, pages 55--59.)

%\bib|lyness1|%
%R.\,C. "Lyness", ``Cycles,'' {\sl The Mathematical Gazette\/ \bf26} (1942),
% 62.

\bib|lyness2|%
R.\,C. "Lyness", ``Cycles,'' {\sl The Mathematical Gazette\/ \bf29} (1945),
231--233.

\bib|lyness3|%
R.\,C. "Lyness", ``Cycles,'' {\sl The Mathematical Gazette\/ \bf45} (1961),
207--209.

\bib|maclaurin|%
Colin "Maclaurin", {\sl Collected Letters}, edited by Stella Mills. \
Shiva Publishing, Nantwich, Cheshire, 1982.

\bib|macmahon|%
P.\,A. "MacMahon", ``Application of a theory of permutations in circular
procession to the theory of numbers,''
{\sl Proceedings of the London Mathematical Society\/ \bf23} (1892), 305--313.

\bib|martzloff|%
J.-C. "Martzloff", {\sl Histoire des Math\'ematiques Chinoises}. \
Paris, 1988.

\bib|mat-ich|%
\t Iu.\,V. Mati{\t\i}asevich, "!Matijasevich"
``Diofantovost' perechislimykh mnozhestv,''
{\sl Doklady Akademii Nauk SSSR\/ \bf191} (1970), 279--282. English translation,
with amendments by the author, ``Enumerable sets are diophantine,''
{\sl Soviet Mathematics\/ \bf11} (1970), 354--357.

\bib|melzak|%
Z.\,A. "Melzak", {\sl Companion to Concrete Mathematics}. \
Volume~1, {\sl Mathematical Techniques and Various Applications}, Wiley,
1973; volume~2, {\sl Mathematical Ideas, Modeling \& Applications}, Wiley, 1976.

\bib|mendelsohn|%
N.\,S. "Mendelsohn", ``Problem E\,2227:
Divisors of binomial coefficients,''
{\sl American Mathematical Monthly\/ \bf78} (1971), 201.

\bib|mersenne|%
Marini "Mersenni", {\sl Cogitata Physico-Mathematica}. \ Paris, 1644.

\bib|mertens-phi|%
F. "Mertens", ``Ueber einige asymptotische Gesetze der Zahlentheorie,''
{\sl Journal f\"ur die reine und angewandte Mathematik\/ \bf77} (1874),
289--338.

\bib|mertens-M|%
"Mertens", ``Ein Beitrag zur analytischen Zahlentheorie,''
{\sl Journal f\"ur die reine und angewandte Mathematik\/ \bf78} (1874),
46--62.

\bib|mills|%
W.\,H. "Mills", ``A prime representing function,''
{\sl Bulletin of the American Mathematical Society}, series~2,
{\bf53} (1947), 604.

\bib|moeb|%
A.\,F. "M\"obius", ``\"Uber eine besondere Art von Umkehrung
der Reihen,''
{\sl Journal f\"ur die reine und angewandte Mathematik\/ \bf9} (1832),
105--123. Reprinted in his {\sl Gesammelte Werke}, volume~4, 589--612.

\bib|moessner|%
A. "Moessner", ``Eine Bemerkung \"uber die Potenzen der nat\"urlichen Zahlen,''
{\sl Sitzungsberichte der Mathematisch\hskip1ptplus1pt-\hskip1ptplus1pt
Naturwissenschaftliche\break Klasse der
Bayerischen Akademie der Wissenschaften}, 1951, Heft~3, 29.{\parfillskip=0pt\par}

\bib|monty-phi|%
Hugh L "Montgomery", ``Fluctuations in the mean of "Euler"'s phi function,''
{\sl Proceedings of the Indian Academy of Sciences}, Mathematical Sciences,
{\bf97} (1987), 239--245.

\bib|montgomery|%
Peter L. "Montgomery", ``Problem E\,2686:
LCM of binomial coefficients,''
{\sl American Mathematical Monthly\/ \bf86} (1979), 131.

\bib|moser-reflections|%
Leo "Moser", ``Problem B-6: Some reflections,''
{\sl Fibonacci Quarterly\/ \bf1},\,4 (1963), 75--76.

\bib|motzkin-straus|%
T.\,S. "Motzkin" and E.\,G. "Straus", ``Some combinatorial extremum problems,''
{\sl Proceedings of the American Mathematical Society\/ \bf7} (1956), 1014--1021.

%\bib|mozzochi|%
%C.\,J. "Mozzochi", ``On the difference between consecutive primes,''
%{\sl Journal of Number Theory\/ \bf24} (1986), 181--187.

\bib|myers|%
B.\,R. "Myers", ``Problem 5795: The spanning trees of an $n$-wheel,''
{\sl American Mathematical Monthly\/ \bf79} (1972), 914--915.

\bib|newton-harm|%
Isaac "Newton", letter to John "Collins" (18~February 1670),
in {\sl The Correspondence of Isaac Newton},
volume~1, 27. % edited by H.\,W. Turnbull Camb U press 1959
Excerpted in {\sl The Mathematical Papers of Isaac Newton},
volume~3, 563. %edited by D.\,T. Whiteside; Camb U press 1969

\bib|niven|%
Ivan "Niven", {\sl Diophantine Approximations}. \
Interscience, 1963.

\bib|niven-gf|%
Ivan "Niven", ``Formal power series,''
{\sl American Mathematical Monthly\/ \bf76} (1969), 871--889.

\bib|odl-wilf|%
Andrew M. "Odlyzko" and Herbert S. "Wilf", ``Functional iteration and the
Josephus problem,'' {\sl Glasgow Mathematical Journal\/ \bf33} (1991),
235--240.

\bib|pascal-digits|%
Blaise "Pascal", ``De numeris multiplicibus,'' presented to Acad\'emie
 Pari\-si\-enne in 1654 and published with his {\sl Trait\'e du
triangle arithm\'etique\/}~\kern-.2pt[|pascal-treatise|].
 Reprinted in {\sl \OE uvres de Blaise Pascal}, volume~3, 314--339.
%\expandafter\write\expandafter\bnx\expandafter{\expandafter[\number\excount]
%\number\pageno.}%

\bib|pascal-treatise|%
Blaise "Pascal", ``Trait\'e du triangle arithmetique,'' in his
{\sl Trait\'e du Triangle Arithmetique, avec quelques autres petits
% sic --- accents are missing in original, including on contents page
% where caps and lowercase are mixed; "trait\'e" has the only accent
traitez sur la mesme matiere}, Paris, 1665. Reprinted
in {\sl \OE uvres de Blaise Pascal\/} (Hachette, 1904--1914), volume~3,
445--503; Latin editions from 1654 in volume~11, 366--390.

\bib|patil|%
G.\,P. "Patil", ``On the evaluation of the negative binomial distribution with
examples,'' {\sl Technometrics\/ \bf2} (1960), 501--505.

\bib|peirce-seq|%
C.\,S. "Peirce", letter to E.\,S. "Holden" (January 1901).
In {\sl The New Elements of Mathematics}, edited by Carolyn "Eisele",
Mouton, The Hague, 1976, volume~1, 247--253. \ (See also page~211.)

\bib|peirce-tree|%
C.\,S. "Peirce", letter to Henry B. "Fine" (17~July 1903).
In {\sl The New Elements of Mathematics}, edited by Carolyn "Eisele",
Mouton, The Hague, 1976, volume~3, 781--784. \ (See also ``Ordinals,''
an unpublished manuscript from circa 1905, in {\sl Collected Papers
of Charles Sanders Peirce}, %edited by Charles "Hartshorne" and Paul "Weiss",
volume~4, 268--280.)

\bib|penney|%
Walter "Penney", ``Problem 95: Penney-Ante,''
{\sl Journal of Recreational Mathematics\/ \bf7} (1974), 321.

\bib|percus|%
J.\,K. "Percus", {\sl Combinatorial Methods}. \ Springer-Verlag, 1971.

\bib|petk|%
Marko "Petkov\v{s}ek", ``Hypergeometric solutions of linear recurrences
with polynomial coefficients,'' {\sl Journal of Symbolic Computation\/ \bf14}
(1992), 243--264.

\bib|pfaff|%
J.\,F. "Pfaff", ``Observationes analytic\ae\ ad {\it L. Euleri\/} institutiones
calculi integralis, Vol.~IV, Supplem.~II \&~IV,''
{\sl Nova acta academi{\ae} scientiarum imperialis Petropolitan{\ae}\/ \bf11},
Histoire section, 37--57. \ (This volume, printed in 1798, contains
mostly proceedings from 1793, although Pfaff's memoir was actually
received in 1797.)

\bib|pochhammer|%
L. "Pochhammer", ``Ueber hypergeometrische Functionen $n^{\rm ter}$
Ordnung,'' 
{\sl Journal f\"ur die reine und angewandte Mathematik\/ \bf71} (1870),
316--352.

\bib|poincare|%
H. "Poincar\'e", ``Sur les fonctions \`a espaces lacunaires,''
{\sl American Journal of Mathematics\/ \bf14} (1892), 201--221.

\bib|poisson|%
S.\,D. "Poisson", ``M\'emoire sur le calcul num\'erique des int\'egrales
d\'efinies,'' {\sl M\'emoires de l'Acad\'emie Royale des Sciences de
l'Institut de France}, series~2, {\bf6} (1823), 571--602.

\bib|polya-counting|%
G. "P\'olya", \kern-2.51pt``Kombinatorische Anzahlbestimmungen f\"ur Gruppen,
Graphen und chemische Verbindungen,'' {\sl Acta Mathematica\/ \bf68}
(1937), 145--254. English translation, with commentary by Ronald~C. "Read",
{\sl Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds},
Springer-Verlag, 1987.

\bib|polya|%
George "P\'olya", {\sl Induction and Analogy in Mathematics}. \
Princeton University Press, 1954.

\bib|polya-pictures|%
G. "P\'olya", ``On picture-writing,''
{\sl American Mathematical Monthly\/ \bf63} (1956), 689--697.

\bib|polya-szego|%
G. "P\'olya" and G. "Szeg\"o", % [sic] from original title page
 {\sl Aufgaben und Lehrs\"atze
aus der Analy\-sis}, two volumes. \ Julius Springer, Berlin, 1925;
fourth edition, 1970 and 1971.
English translation, {\sl Problems and Theorems in Analysis}, 1972 and~1976.

%\bib|poonen|%
%Bjorn "Poonen", ``Josephus sets.''
%Unpublished manuscript, 1987.

%\bib|qian|%
%"Qi\'an" B\u{a}o-Zh\=ong, {\sl Zh\=ong-G\'uo Sh\`u X\'ue Sh\u{\i}\/} [Chinese
%mathematics history]. \ Beijing, 1964. [Sometimes romanized as
%"Ch'ien" Pao-Ts'ung, {\sl Chung-Kuo She Hs\"ueh Shih}.]

\bib|rado|%
R. "Rado", ``A note on the Bernoullian numbers,''
{\sl Journal of the London Mathematical Society\/ \bf9} (1934), 88--90.

\bib|rainville|%
Earl D. "Rainville", ``The contiguous function relations for $_pF_q$ with
applications to "Bateman"'s $J_n^{u,v}$ and "Rice"'s $H_n(\zeta,p,v)$,''
{\sl Bulletin of the American Mathematical Society}, series~2,
{\bf51} (1945), 714--723.

\bib|raney|%
George N. "Raney", ``Functional composition patterns and power series
reversion,'' {\sl Transactions of the American Mathematical Society\/ \bf94}
(1960), 441--451.

\bib|d-r-rao|%
D. Rameswar "Rao", ``Problem E\,2208: A divisibility problem,''
{\sl American Mathematical Monthly\/ \bf78} (1971), 78--79.

\bib|rayleigh|%
John William "Strutt", Third Baron "Rayleigh", {\sl The Theory of Sound}. \
First edition, 1877; second edition, 1894. \ (The cited material about
irrational spectra is from section 92a of the second edition.)

\bib|recorde-wit|%
Robert "Recorde", {\sl The Whetstone of Witte}. \
London, 1557.

\bib|asymptotic-query|%
Simeon "Reich", ``Problem 6056: Truncated exponential-type series,''
{\sl American Mathematical Monthly\/ \bf84} (1977), 494--495.

\bib|rham|%
Georges de "Rham", ``Un peu de math\'ematiques \`a propos d'une courbe plane,''
{\sl Elemente der Mathematik\/ \bf2} (1947), 73--76, 89--97.
Reprinted in his {\sl \OE uvres Math\'ematiques}, 678--689.

\bib|ribenboim|%
Paolo "Ribenboim", {\sl 13 Lectures on Fermat's Last Theorem}. \
Springer-Verlag, 1979.

\bib|riemann-sums|%
Bernhard "Riemann", ``Ueber die Darstellbarkeit einer Function durch eine
trigonometrische Reihe,'' Habilitations\-schrift, G\"ottingen, 1854.
Published in {\sl Abhandlungen der mathematischen Classe der
K\"oniglichen Gesellschaft der Wissenschaften zu G\"ottingen\/ \bf13}
(1868), 87--132. Reprinted in his {\sl Gesammelte Mathematische Werke},
227--264.

\bib|roberts|%
Samuel "Roberts", ``On the figures formed by the intercepts of a system
of straight lines in a plane, and on analogous relations in space of
three dimensions,''
{\sl Proceedings of the London Mathematical Society\/ \bf19} (1889), 405--422.

\bib|roedseth|%
\O ystein "R\o dseth", ``Problem E\,2273:
Telescoping Vandermonde convolutions,''
{\sl American Mathematical Monthly\/ \bf79} (1972), 88--89.

\bib|rosser-schoenfeld|%
J. Barkley "Rosser" and Lowell "Schoenfeld", ``Approximate formulas for
some functions of prime numbers,'' {\sl Illinois Journal of Mathematics\/
\bf 6} (1962), 64--94.

\bib|rota-mobius|%
Gian-Carlo "Rota", ``On the foundations of combinatorial theory.~I.~
Theory of M\"obius functions,'' {\sl Zeitschrift f\"ur
Wahrscheinlichkeitstheorie und verwandte Gebiete\/ \bf2} (1964), 340--368.

\bib|roy|%
Ranjan "Roy", ``Binomial identities and hypergeometric series,''
{\sl American Mathematical Monthly\/ \bf94} (1987), 36--46.

\bib|saalschutz|%
Louis "Saalsch\"utz", ``Eine Summationsformel,''
{\sl Zeitschrift f\"ur Mathematik und Physik\/ \bf35} (1890), 186--188.

\bib|salty-phi|%
A.\,I. "Saltykov", ``O funktsii \'E\u{\i}lera,''
{\sl Vestnik Moskovskogo Universiteta}, series 1, Matematika, Mekhanika
(1960), number~6, 34-