Pacific Journal of Mathematics

Transcription

1 Pacific Journal of Mathematics SOME CONGRUENCES FOR THE BELL POLYNOMIALS L. CARLITZ Vol. 11 No

2 SOME CONGRUENCES FOR THE BELL POLYNOMIMALS L. CARLITZ 1. Let a lf a 2 a 3 denote indeterminates. The Bell polynomial φ n (a lf a 2 a 3 ) may be defined by φ 0 = 1 and (1.1) Φn = *(««α. ) = Σ ^ Σ where the summation is over all nonnegative integers k ά such that k λ + 2k 2 + 3fc 3 + == n. For references see Bell [2] and Riordan [5 p. 36]. The general coefficient (1.2) A n { h K k t ) = fc J is integral; this is evident from the representation )! (3fe)! and the fact that the quotient (rk)l kl(r\y is integral [1 p. 57] The coefficient A n (k lf k 2 fc 3 ) resembles the multinomial coefficient M(k lf K fc s ) = ~ k + & + fc + )! k λ \k 2 \kz If p is a fixed prime it is known [3] that M(k 19 k 2 k 3 ) is prime to p if and only if ki = Σ a ijp j (0 ^ α iy < p) and &! + fc 2 + fc 3 + = Σ ajp j (0 ^ α y < p) i It does not seem easy to find an analogous result for A n (k lf k 2 k z For some special results see 3 below. ). Received October

3 1216 L. CARLITZ Bell [2] showed that (1.3) φ p = a» + a p (modi?) and also determined the residues (mod p) of φ p+1 φ p+2 φ p+3. He also obtained an expression for the residue of φ p+r as a determinant of order r + 1. Generalizing (1.3) we shall show first that (1.4) φ p r = af + cΰγ 1 + +a pr (mod p) and that (1.5) φpnitti a 2J α 3..) = φ n (φ p ατ 21> a w ) (mod p) for all n ^ 1. Note that on the right the first argument in φ n is φ p and not tf p. (2.1) 2. From (1.1) we get the generating function Indeed this may be taken as the definition of φ n. respect to t we get Differentiating with oo J. n oo J. n oo J.r Σ Φ» +1 -V - Σ Φn-tγ Σ «r+1 -V ' «=0 721 n=0 ^! r=0 r\ so that (2.2) ψ M+1 = Since the binomial coefficient unless p r and it follows from (2.2) that (2.3) φ pn+1 = Σ (Jf)φ ( _ r) α w+x (mod p) If for brevity we put A(t) = Σ cc r tηrl l

4 SOME CONGRUENCES FOR THE BELL POLYNOMIALS 1217 SO that n=0 n\ it is easily seen by repeated differentiation and by (1.3) that (2.4) Σ Φ*+*%f ~ WW)* + A (t)}e AW (mod p). (By the statement Σ ^n r = Σ B n r (mod m) w=o 72Ί Λ=O 72<! where A n B n are polynomials with integral coefficients is meant the system of congruences A n = B n (mod ra) (n = )). Hurwitz [4 p. 345] has proved the lemma that if a lt α 2 α 3 arbitrary integers then are The proof holds without change when the a n are indeterminates. Since A\t) = Σ α. + rζit follows easily from Hurwitz's lemma that Thus (2.4) becomes which yields (A\tγ - (a x + Σ ^ + 1^-) P = αf (mod p). Σ Φn+i.-V = («ϊ + Σ «r+ -V) Σ Φ»-V ' w=o 721 \ r=o 7*J / Λ=o 721 (2.5) φ n+p = (α? + α p )φ Λ + Σ (J^απ-A-. (mod p). In particular for n = 0 (2.5) reduces to Bell's congruence (1.3). Similarly ΦP+I = (α? + a p )a x + α p+1 Ξ φ p a λ + α p+1 Φ P+a = ψ p φ 2 + 2a p+1 a x + a p+2

5 1218 L. CARLITZ and so on. We remark that (2.5) is equivalent to Bell's congruence involving a determinant [2 p. 267 formula (6.5)]. Also for s = a x = a 2 (2.5) reduces to (2.5)' a n+p (s) = (s ± p + s)a n (s) + s where [5 p. 76] = a n+1 (s) + s*a n (s) (mod p) The con- and S(n k) denotes the Stirling number of the second kind. gruence (2.5)' is due to Touchard [6] If in (2.5) we replace n by pn we get n ίn\ (2.6) Φpίn + l) = ΦpΦnp + Σ ( r )<Xp{r+l)Φpln-r) (mθd p) for all n = Thus φ pn is congruent to a polynomial in φ p (Xϊpy <*3 P alone. Moreover comparing (2.6) with (2.2) it is clear that (2.7) φ pn = φ n {φ py a 2pj a p...) (mod p) so that we have proved (1.5). Replacing n by pn in (2.7) we get Φp*n = ΦpniΦp &2p M 3p ) = φ n (φ* + Cί p <L (X 2p 2 (X& ). In particular for n = 1 Again replacing ^ by p^ we get a% so that in particular ΦΛ = φ w (Φ^2 Continuing in this way we see that af (2.8) φ p r n = φ n (Φ P r a 2p r a p r. ) (mod p) and (2.9) φ p r Ξ φ p l_ x + a p r = af + a* r ~~ <v (mod p). We have therefore proved (1.4) as well as the more general congruence (2.8).

6 SOME CONGRUENCES FOR THE BELL POLYNOMIALS 1219 Since φ 2 = a\ + a 2 φ s = a\ + Za x a 2 + a 3 φ 4 = oίί + a\a 2 + Aa x a z + Sal + a it follows from (2.8) that! 'Φ*pr = ΦJr + OΓ 2p r φ 3p r = φjr + 3φ p r0: 2l) r + α 8p r 04p» = Φ P r + 6φjrαr 2p r + 4φ p rα 3p r + 3aτj p r + α: 42 r and so on. We note also that (2.3) implies (2.11) ί! + Sφ 2p r(x p r +1 3 By means of (1.5) we can obtain certain congruences for the coefficient A(k 19 k 2f k 39 ) Indeed by (1.1) and (1.3) (3.1) φ n {φ pj a 2p a 3pj...) = Σ A n {k ly K K ' OK + a p )^a\ιa\ι... (mod p) where the summation is over nonnegative fc y such that k x + 2fc 2 + 3fc 3 + = n. The right member of (3.1) is equal to (Ί 9\ V A (h h h... 1 V I λ \fy p (kj) r=0 \' On the other hand (3.3) φ pn = Σ Apnihi h 2 h t summed over (3.4) K + 2h 2 + 3h z + = p n. It follows from (1.5) that A pn (h 19 h 2 h 3 ) = 0 (mod p) except possibly when (3.5) hj = 0 (j>i fp + j) m When this condition is satisfied (3.4) becomes

7 1220 L. CARLITZ K + p(h p + 2h 2p + ) = pn consequently h x pk λ and (3.3) becomes Φ Pn = Σ A pn (pk lf h 9.)a We have therefore proved the following result: THEOREM 1. The coefficient A pn (h lf h 2 h 3 ) occurring in (3.3) is certainly divisible by p unless (3.5) is satisfied and h x = pk γ. If these conditions are satisfied then ίk \ A pn (h lf h 2 h Zi ) = ( * )A n {k 1 h p h p h 2p ) (mod p). \' i/ p / If we make use of (1.4) we obtain the following simpler THEOREM 2. Let h x + 2h 2 + Sh 3 + = p r. Then the coefficient A p r(h lf h 2 h 3 «) is divisible by p except when for some j y in which case h = 0 (iφ o) hj = p s A p r(h 19 h 2J h z ) = 1 (mod p). Using (2.10) and (2.11) we can obtain additional results. For example take h x + 2h 2 + 3& 3 + = 2p r. Then A 2p r{h lf h 2 h 3 ) is divisible by p unless (i) all h s = 0 (s Φ j) hj = 1 or 2; (ii) all h s = 0 (s Φ i j) hi = hj = 1. In case (i) A Ξ 1 in case (ii) 4Ξ2 (mod p). For ^ = Sp r the corresponding results are more complicated. 4 We turn now to the polynomial C n (a lf a 2 a z ) the cycle indicator of the symmetric group [5 p. 68]: (4.1) C n = C n (a lf a 2y a 3 ) = φ n (a 19 a 2 2la z ) ^ k λ \k 2 \k z... \ l / \ 2 / \ 3 where the summation is over all nonnegative fey such that k λ + 2k 2 + 3fc 3 + = n. It is convenient to define C o = 1.

8 SOME CONGRUENCES FOR THE BELL POLYNOMIALS 1221 Put the general coefficient of C H. Clearly c n (k lf k 2 k 3 ) is integral and indeed a multiple of A n (k 19 k 2 k 3 ). From (4.1) we get the generating function (4.3) G(t) = Σ <? = exp (a x t + α 2 ί 2 + α 3 ί 8 + n=0 tl! V 2 3 For brevity put Differentiating (4.3) with respect to t we get that is G'(t) - C'( oo j.<n oo oo j.n w=0 72J r=0 Λ=0 7l! This implies (4.4) u w+1-2 so that (4.5) C n+1 = a λ C n (mod n). By repeated differentiation of (4.3) we get (compare (2.4)) (4.6) -^-G(ί) = {(C'(t)) p + C (p) (ί)}g(ί) (mod p). Now since C f (ί) = ^P cc t n C ^pht) y 1 ίti 4- x) lvct n=0 n=0 92 it is clear that (C'(ί))'sαϊ C^ίίjs-α (modp); at the last step we have used Wilson's theorem. Thus (4.6) becomes

9 1222 L. CARLITZ so that Σ C n+p^- = (αϊ - a) Σ w=o n\ «=o nl cλ- (4.7) C n+p = (α? - α p )C n (mod p). In particular we have (4.8) C p = a\ - a p (mod p) and (4.9) C n+rp = (α? - α p ) r C n (mod p) We remark that for p = (4.8) is in agreement with the explicit values of C n given in [5 p. 69]. By (4.9) with n = 0 we find that the coefficient c r (fci fc 2 & 3 ) Ξ 0 (mod p) unless all fc except k λ and fc p vanish and k L is a multiple of p; in this case we have (4.10) c rp (pk 0 0 k p ) = (-1)*'( ) (mod p).. REFERENCES 1. P. Bachmann Niedere Zahlentheorie vol. 1 Leipzig E. T. Bell Exponential polynomials y Annals of Math. 35 (1934) L. E. Dickson Theorems on the residues of multinomial coefficients with respect to a prime modulus Quarterly J. Math. 33 (1902) A. Hurwitz ϋber die Entwickelungscoeffizienten der lemniscatischen Functionen Mathematische Werke Basel 2 (1933) J. Riordan An Introduction to combinatorial analysis New York J. Touchard Proprietes arithmetiques de certains nombres recurrents Ann. Soc. Sci. Bruxelles A53 (1933) DUKE UNIVERSITY

10 PACIFIC JOURNAL OF MATHEMATICS EDITORS RALPH S. PHILLIPS A. L. WHITEMAN Stanford University University of Southern California Stanford California Los Angeles 7 California F. H. BROWNELL L. J. PAIGE University of Washington University of California Seattle 5 Washington Los Angeles 24 California ASSOCIATE EDITORS E. F. BECKENBACH D. DERRY H. L. ROYDEN E. G. STRAUS T. M. CHERRY M. OHTSUKA E. SPANIER F. WOLF SUPPORTING INSTITUTIONS UNIVERSITY OF BRITISH COLUMBIA STANFORD UNIVERSITY CALIFORNIA INSTITUTE OF TECHNOLOGY UNIVERSITY OF TOKYO UNIVERSITY OF CALIFORNIA UNIVERSITY OF UTAH MONTANA STATE UNIVERSITY WASHINGTON STATE COLLEGE UNIVERSITY OF NEVADA UNIVERSITY OF WASHINGTON NEW MEXICO STATE UNIVERSITY * * * OREGON STATE COLLEGE AMERICAN MATHEMATICAL SOCIETY UNIVERSITY OF OREGON CALIFORNIA RESEARCH CORPORATION OSAKA UNIVERSITY HUGHES AIRCRAFT COMPANY UNIVERSITY OF SOUTHERN CALIFORNIA SPACE TECHNOLOGY LABORATORIES NAVAL ORDNANCE TEST STATION Mathematical papers intended for publication in the Pacific Journal of Mathematics should be typewritten (double spaced) and the author should keep a complete copy. Manuscripts may be sent to any one of the four editors. All other communications to the editors should be addressed to the managing editor L. J. Paige at the University of California Los Angeles 24 California. 50 reprints per author of each article are furnished free of charge; additional copies may be obtained at cost in multiples of 50. The Pacific Journal of Mathematics is published quarterly in March June September and December. The price per volume (4 numbers) is $12.00; single issues $3.50. Back numbers are available. Special price to individual faculty members of supporting institutions and to individual members of the American Mathematical Society: $4.00 per volume; single issues $1.25. Subscriptions orders for back numbers and changes of address should be sent to Pacific Journal of Mathematics 103 Highland Boulevard Berkeley 8 California. Printed at Kokusai Bunken Insatsusha (International Academic Printing Co. Ltd.) No. 6 2-chome Fujimi-cho Chiyoda-ku Tokyo Japan. PUBLISHED BY PACIFIC JOURNAL OF MATHEMATICS A NON-PROFIT CORPORATION The Supporting Institutions listed above contribute to the cost of publication of this Journal but they are not owners or publishers and have no responsibility for its content or policies. Reprinted 1966 in the United States of America

11 Pacific Journal of Mathematics Vol. 11 No A. V. Balakrishnan Prediction theory for Markoff processes Dallas O. Banks Upper bounds for the eigenvalues of some vibrating systems A. Białynicki-Birula On the field of rational functions of algebraic groups Thomas Andrew Brown Simple paths on convex polyhedra L. Carlitz Some congruences for the Bell polynomials Paul Civin Extensions of homomorphisms Paul Joseph Cohen and Milton Lees Asymptotic decay of solutions of differential inequalities István Fáry Self-intersection of a sphere on a complex quadric Walter Feit and John Griggs Thompson Groups which have a faithful representation of degree less than (p 1/2) William James Firey Mean cross-section measures of harmonic means of convex bodies Avner Friedman The wave equation for differential forms Bernard Russel Gelbaum and Jesus Gil De Lamadrid Bases of tensor products of Banach spaces Ronald Kay Getoor Infinitely divisible probabilities on the hyperbolic plane Basil Gordon Sequences in groups with distinct partial products Magnus R. Hestenes Relative self-adjoint operators in Hilbert space Fu Cheng Hsiang On a theorem of Fejér John McCormick Irwin and Elbert A. Walker On N-high subgroups of Abelian groups John McCormick Irwin High subgroups of Abelian torsion groups R. E. Johnson Quotient rings of rings with zero singular ideal David G. Kendall and John Leonard Mott The asymptotic distribution of the time-to-escape for comets strongly bound to the solar system Kurt Kreith The spectrum of singular self-adjoint elliptic operators Lionello Lombardi The semicontinuity of the most general integral of the calculus of variations in non-parametric form Albert W. Marshall and Ingram Olkin Game theoretic proof that Chebyshev inequalities are sharp Wallace Smith Martindale III Primitive algebras with involution William H. Mills Decomposition of holomorphs James Donald Monk On the representation theory for cylindric algebras Shu-Teh Chen Moy A note on generalizations of Shannon-McMillan theorem Donald Earl Myers An imbedding space for Schwartz distributions John R. Myhill Category methods in recursion theory Paul Adrian Nickel On extremal properties for annular radial and circular slit mappings of bordered Riemann surfaces Edward Scott O Keefe Primal clusters of two-element algebras Nelson Onuchic Applications of the topological method of Ważewski to certain problems of asymptotic behavior in ordinary differential equations Peter Perkins A theorem on regular matrices Clinton M. Petty Centroid surfaces Charles Andrew Swanson Asymptotic estimates for limit circle problems Robert James Thompson On essential absolute continuity Harold H. Johnson Correction to "Terminating prolongation procedures"