Stern Vault

On this page, you will find links to various Stern-related papers.

Week One

Classic papers

  1. Stern : M. A. Stern, Ueber eine zahlentheoretische Funktion, J. Reine Angew. Math., 55 (1858), 193--220. (German).
  2. Lehmer: D. H. Lehmer, On Stern's diatomic series, Amer. Math. Monthly, 36 (1929), 59--67.

Local authors

  1. Dilcher and Stolarsky, 1 : K. Dilcher and K. B. Stolarsky, A polynomial analogue to the Stern sequence. Int. J. Number Theory 3 (2007), no. 1, 85–103.
  2. Reznick 1: B. Reznick, Regularity properties of the Stern enumeration of the rationals. J. Integer Seq. 11 (2008), no. 4, Article 08.4.1, 17 pp., arXiv -- math.NT/0610601.
  3. Anders, Dennison, Lansing and Reznick: K. Anders, M. Dennison, J. W. Lansing and B. Reznick, Congruence properties of binary partition functions, to appear in Ann. Combin.

Historical background

  1. Hayes : B. Hayes, On the teeth of wheels, The American Scientist, July-August 2000, Volume 88, Number 4, p. 296.
  2. Rowe : D. Rowe, "Jewish Mathematics" at Gottingen in the Era of Felix Klein, Isis, Vol. 77, No. 3 (Sep., 1986), pp. 422-449.

Week Two

Classic papers (But see week 6)

  1. Our scanner decided to make separate .pdf's for each page of this article by G. Halphen from Bull. Soc. Math Paris, vol. V, 1877: Page one, Page two, Page three, Page four, Page five, Page six: (French).
  2. Commentary by Eduard Lucas on Halphen's paper, in the same journal, same year, same language.
  3. Minkowski : H. Minkowski, Zur Geometrie der Zahlen, talk at 1904 International Congress of Mathematicians (German).
  4. According to this 2007 blog post, Minkowski may have given the first talk ever using slides. The article reproduces several of the slides.
  5. Hancock : H. Hancock, Development of the Minkowski Geometry of Numbers, Macmillan, 1939. In the 1930s, Hancock wrote two volumes summarizing and describing Minkowski's papers on the subject; this is what he said about the ? function.

Local authors

  1. Dilcher and Stolarsky : K. Dilcher and K. B. Stolarsky,. Stern polynomials and double-limit continued fractions. Acta Arith. 140 (2009), 119–134. (Linked are galley-proofs).

Stern, but not by name

  1. Williams and Browne: G. T. Williams and D. H. Browne, A family of integers and a theorem on circles, Amer. Math. Monthly, 54 (1947), 534--536.
  2. In two letters, one and two, the great computer scientis E. W. Dijkstra rediscovers the Stern sequence and several properties discussed in the first chapter. He only mentions other appearances at the end. This is taken from: E. W. Dijkstra, Selected Writings on Computing: A Personal Perspective, Springer-Verlag, 1982.

Week Three

New papers from the arXiv

  1. Allouche and Mendes France: J.-P. Allouche and M. Mendes France, Stern-Brocot polynomials and power series, preprint.
  2. Coons and Shallit: M. Coons and J. Shallit, A pattern sequence approach to Stern's sequence, preprint.

From the Monthly

  1. Calkin and and Wilf: N. Calkin and H. Wilf, Recounting the Rationals, American Mathematical Monthly, 107 (2000), 360-- 363.
  2. Benito and Escribano: M. Benito and J. Escribano, An easy proof of Hurwitz' Theorem, American Mathematical Monthly, 109 (2002), 916--918. [Uses the Brocot array] .
  3. Knuth: A monthly problem by D. Knuth with a different definition of t(n). Rationals, American Mathematical Monthly, 110 (2003), 642-643.

Squashing partitions

  1. Sloane and Sellers: On non-squashing partitions, Discrete Mathematics, 294 (2005), 259--274.

Historical background

  1. Stern in 1906 : The entry on Stern in the 1906 reference work: "The Jewish Encyclopedia".

Local authors

  1. Reznick2: B. Reznick, Some extremal problems for continued fractions, Illinois J. Math, 29 (1985), 261--279. [My first reference to Sternstuff (pp.270-271); reference [15] has become the classnotes.]

Week Five

Classic papers (though not directly Stern)

  1. Guy1: R. Guy, The strong law of small numbers, American Mathematical Monthly, 95 (1988), pp. 697-712.
  2. Guy2: R. Guy, The second strong law of small numbers, Mathematics Magazine, 63 (1990), pp. 3-20. (To both: "Nice looking sequence you've got starting there. Think it will continue? In the immortal words of Clint Eastwood, you've got to ask yourself one question: 'Do I feel lucky?' Well, do ya, grad student?")

Recent papers, largely from "Integers" (on-line, so no page numbers)

  1. Chan: S. H. Chan, Analogs of the Stern sequence, Integers, 11 (2011) #A26. (Song is a 2005 student of Bruce Berndt's.)
  2. Bacher: R. Bacher, Twisting the Stern sequence, arXiv:1005.5627. (Probably will not be published.)
  3. Coons1: M. Coons, On some conjectures concerning Stern's sequence and its twist, Integers, 11 (2011) #A35.
  4. Coons2: M. Coons, A correlation identity for Stern's sequence, 12 (2012) #A3.

Week Six

More on twisting the Stern sequence

  1. Allouche: J.-P. Allouche, On the Stern sequence and its twisted version, arXiv 1202.417v1.

Better .pdfs of the 19th century French papers from Week 2

(Tip o' the chapeau to Jean-Paul Allouche )

  1. Halphen
  2. Lucas:

Local authors (that is, references for section 3.8 and 4.1)

  1. Reznick3: B. Reznick, On the length of binary forms, to appear in Quadratic and Higher Degree Forms , (K. Alladi, M. Bhargava, D. Savitt, P. Tiep, eds.), Developments in Math. Springer, New York,
  2. Reznick4: B. Reznick, Sums of even powers of real linear forms, Mem. Amer. Math. Soc., Volume 96, Number 463, March, 1992 (MR 93h.11043). Webpage contains links to each chapter, separately scanned and posted.
  3. Reznick5: B. Reznick, Continued fractions and an annelidic pde, Math. Intelligencer, 5 (1983), 61-63. (MR 86e.11007). Written while waiting for tenure to be approved. Never cited anywhere else. Only paper in MathSciNet to use the word "annelidic"