2l WEIGHTED STIRLING NUMBERS OF THE FIRST AND SECOND KIND II [Oct. 6. CONCLUSION We have proven a number-theoretical problem about a sequence, which is a computer-oriented type, but cannot be solved by any computer approach. REFERENCE. J. N i e v e r g e l t, J. C. F a r r a r, & E. M. Reingold. Computer Approaches to Mathematical Problems. New J e r s e y : P r e n t i c e - H a l l, 74. Ch. 5... WEIGHTED STIRLING NUMBERS OF THE FIRST AND SECOND KIND II L. CARLITZ Due University, Durham, NC 2776. INTRODUCTION The Stirling numbers of the first second ind can be defined by (.) () n = ( + ) ( + n - ) = ] T 5 (n, ) 9 j n (.2) n = ]T S(n 9 ).( - ) ( - + ), = o respectively. In [6], the writer has defined Weighted Stirling numbers of the first second ind, S (n 9 9 A) S(n 9 9 A ), by maing use of certain combinatorial properties of S (n 9 ) S(n 9 ). Numerous properties of the generalized quantities were obtained. The results are somewhat simpler for the related functions: ( R (n 9 9 A) =~S {n 9 +, X) + S (n 9 ) (.) I { R(n 9 9 X) = S(n 9 +, A) + S(n 9 ). In particular, the latter satisfy the recurrences, (.4) = R (n 9 s X) = R (n 9 -, X) + (n + A)i? (n, fc, A) the orthogonality relations (.5) i?(n, fe, A) = R(n 9 -, A) + ( + A)i?(n, fc, A ), n /?(n,, A) (-iy R (.j,, X) We have also the generating functions {~}) n -R {n,, X)RU,, A) = {J (" J. j = d.6) J T E ^ I ^ ni n = li = > ) y = ( - y \-y
98] WEIGHTED STIRLING NUMBERS OF THE FIRST AND SECOND KIND II 24 (.7) Yl f l & f a n= the eplicit formula n - > ^ = e X ep{2/(e* - )}, (-8) R(n,, A) = ^ r E (-D " J (5)w + * ) " Moreover, corresponding to (.) (.2), we have n (-9) (X + z/) n = i? ( n, > \)y 7< = n (-) y n = (-l) n "*i?(n,.fc, X)(z/ + \). = It is well nown that the numbers S (n 9 n-) 9 S(n 5 n-) are polynomials in n of degree 2, In [4] it is proved that S (.n, n - ) = > <fc, J)( n + / fe X ) (.) < j=± i - S(n 9 n v=±b(,d n+ i - i \ - where B ( 9 j) s B( 9 j) are independent of n 9 X (>l) 9 (.2) B^, J) = S( 9 - j + ), ( <. J <. fc). The representations (.) are applied in [4] to give new proofs of the nown relations (.) \ S(n, n-) - ( * ")( I J)M* + t, t) S ln.n-k) =ib( + _i)( lty( + t, t). For references to (.), see [2], [7]. One of the principal objectives of the present paper is to generalize (.). The generalized functions R (n 9 n - 9 \) 9 R(n 9 n- 9 X) are also polynomials in n of degree 2. We show that R^n, n - 9 X) = X X (, J\ X > ( n 2 V ) = ^ (.4) < / p \ R{n 9 n - 9 X) = X B ^ ^» X) l 2 J j - o ^ where B { 9 J, X), B(fc, j, X) are independent of n 9 (.5) B ( 9 j 9 X) = B( 9 - j, - X), ( <_ j < ).
2h WEIGHTED STIRLING -NUMBERS OF THE FIRST AND SECOND KIND I I [Oct. (-6), As an application of (.4) (.5), it is proved that i lis JL <*, J_ \ / Z> _ \ *(. n -, X) = (* + % )(* I I " t >(^ + *.*.! ^ ( n, n - fc, X) = X ) (^ t " t L )(* fe + I X ) R (- + t, t, - \ ). t -o ^ v For X =, (.6) reduces to (.) with n replaced by n + ; for X =, we apparently get new results. In the net place, we show that (/?(«, n -, X) = (l)b ( - n+) (X) (.7) < (/?(«, n - t W = f " J " ^ " ^ ( l - X), (7c) where B (X) is the Bernoulli polynomial of higher order defined by [8 9 Ch. 6]: IX^fH^p"- We remar that (.7) can be used to give a simple proof of (.6). For the special case of Stirling numbers, see [2], It is easily verified that 9 for X =, (.7) reduces to wellnown representations [8, Ch. 6] of S(n 9 n - ) S (n s n - ). In view of the formulas (for notation references see []), fc-i A) (.8) \ \ i t is of interest to define coefficients R f ( s j $ X) R^( s j s X) by means R(n 9 n - fe, A) = > f ( s J, ^)( 2 fc W - j ) (.9) ^ \ = ^ ( n, n - 9 X) - ^ ^ ( f c, j, X) (^2 \ j - o Each coefficient Is a polynomial in X of degree 2 has properties generalizing those of S f ( 9 j) S*( s j ). Finally ( 9) 9 we derive a number of relations similar to (.6), connecting the various functions defined above. For eample, we have RAn, n -, X) = V =t(-d (-)""- -\ n + 7 " +{)R>V<, T i-w(, - j, - X) (.2) \ *l R{n, n -, X) = ]T (-l)*"^ * ^(fc, fe - J, - A) ^j.
98] WEIGHTED STIRLING NUMBERS OF THE FIRST AND SECOND KIND II 25 / A / _ +\ R[(n 9 9 X) = (^Wfe _ tv? ( n 9 t s * " (.2) < V X) In the proofs s we mae use of the relations (.5). 2. REPRESENTATIONS OF R(n, n - 9 \) As a special case of a more general result proved in [5], if f() is an arbitrary polynomial of degree <m 9 then there is a unique representation in the form m-l /, «\ (2.D /(*> = >, m \ J = where the a are independent of» Thus s since R(n s n - 9 X) is a polynomial in n of degree 2 9 we may put 9 for _> l s X (2.2) R(n 9 n - fc, X) = ^S(?c } j\ M* z)> where the coefficients 5(fc 9 j, X) are independent of n, By (.4) 5 we have 9 for >, (2.) R(n +, n - +, X) = (n - & + + X)i?(n, n - fc +, X) Thus, (2.2) y i e l d s + R(n 9 n - 9 X). 2,. \ 2-2 >(fc, j, X) 2V- J l) = <«- * + + *> 2>(fc -, J, X)g, _ J 2). J. - - ^ J - ^ Since n - fe + + X = (n + j - 2 + 2) + (fe - j - + X), we get I t follows t h a t + J (2.4) S( 9 j 9 A) = ( + j-\)b(-l 9 j 9 X)+ (-j+\)b(-, j -, X). We shall now compute the first few values of B( 9 j 5 X). To begin with we have the following values of R(n 9 n - 9 X). Clearly, R(n 9 n 9 X) =. Then, by (2.), with =, we have It follows that R(n +, n 9 X) - R(n 9 n - \ 9 X) = n + X. (2.5) i?(n, w -, X) (»)+».
246 WEIGHTED STIRLING NUMBERS OF THE FIRST AND SECOND KIND II [Oct. Net, taing = 2 in (2.), R(n +, n -, X) - R(n 9 n - 2, X) we find that (2.6) R(n 9 n - 2, X) (n - + X)i?(n, n -, X), (fl+ghg)** *- <«^>- A little computation gives the following table of values: Bi, j, A) > < 2 - X X 2 ( - X ) 2 + X - 2X 2 X 2 ( - X ) 8 + 7X - 2X 2 + X + 4X + 6X 2 - X X The last line was computed by using the recurrence (2.4). Note that the sum of the entries in each row above is independent of X. This is in fact true generally. By (2.2), this is equivalent to saying that the coefficient of the highest power of X in R(n 9 n - 9 X) is independent of X. To prove this, put R(n, n - 9 X) = a n 2 + an 2^" +. Then R(n +, n - +, X) - R(n 9 n - 9 X) Thus, by (2.), 2a. Therefore, (2.7) = a ((n + l) 2 - n 2 ) + a fe ((n + l ) 2 * " - n 2 ^ ) +... = 2a n 2 +.... Y,B{ 9 j, X) j=o z _. Since a = %, we get ^ = y- 2(2-2)... 2 2 l (2) \ 2 \..5... (2 - ). This can also be proved by induction using (2.4). However, the significant result implied by the table together with the recurrence (2.4) is that (2.8) B( 9 J, X) =, (j > ). Hence, (2.2) reduces to (2.9) R(n 9 n - 9 X) = B( 9 j, A)( n ~t J ). It follows from (2.9) that the polynomial < n <. R(n 9 n - 9 X) vanishes for
98] WEIGHTED STIRLING NUMBERS OF THE FIRST AND SECOND KIND II il Incidentally, we have anticipated (2.9) in the upper limit of summation in (2.7).. REPRESENTATION OF R (n, n -, X) Since R (n 9 n - 9 \) is a polynomial in n of degree 7, we may put, for _>, (.) R^n 9 n - 9 X) = X X ^ > J. X) \2) 9 where B ( 9 j, X) is independent of n. By (.4) we have, for >, (.2) ^ ( n + l, n - +, X) = (n + X)i? (n, n-+l 9 X) + i? (n, n - f c, A). Thus, by (. ), we get 2.. 2fe-2 I ^ f t. J. ^ U f c - l - («+ ** *! < * -!» J X > 2" - 2 j - - = n \ j - - \ / + It follows that (.) B^ 9 J, X) = (j + l - X ^ O : -, j, X) + (2-j- l + X)B (-, j -, X ). As in the previous section, we shall compute the first few values of B (» j\ X). To begin with, we have R (n 9 n, X) =. Then by (.2), with =, we have R (n + 5 n 9 X) - R (n, n -, X) = n + X, so that (.4) fl^n, n -, X) = ( 2 ) Net, t a i n g fc = 2 in (. 2 ), I t follows + n i? r (n +, n -, A) - i?!(w, n - 2, X) = (n + X)i? (n, n -, X). t h a t (.5) Vn.»-2.X)-.(j) + 2(5) + { (») + (;)} + (;)S ( n > 2 ). A little computation gives the following table of values:
2^8 WEIGHTED STIRLING NUMBERS OF THE FIRST AND SECOND KIND II [Oct. B ( 9 j, X) \ 2 - X X 2 ( - X) 2 2 + X - 2X 2 a) 2 ( - X) 8-7X - X 2 + X 6 + 8X - X 2 - X <*> Eactly as above, we find that (.6) YjB^, j, X) ="^ 7= -.4... (2 - ). This can also be proved by induction using (.). Moreover, (.7) B ( 9 j, X) =, (j > ), so that (.) becomes (.8) R (n 9 n -, X) = J^B^, * vl? 2^\ Thus, the polynomial R Y (n 9 n -, X) vanishes for <_n <.. RELATION OF B (, j, X) TO B(, j, X) In (2.4) replace j by - j we get (4.) B( 9 - j, X) = (2 - Q - \)B( -, - j, X) Put Then (4.) becomes + (j + \)B( -, - j -, X). _ B( 9 j, X) = B( - j, X ). (4.2) B( 9 j, X) = (2 - j - X)B( -, j -, X) Comparison of (4.2) with (.) gives therefore + (j + \)B( -, j, X). i.(fc> J, A) = B(, J, - X), (4.) B ( 9 j, X) = B( 9 - j, - X). (4.4) In particular, We recall that B ( 9, X) = B( 9 9 - X) = ( - X) ^(, 9 X) = 5(,, - X) = (X). (4.5) R(n 9, 9 ) = 5(n, ), R(n 9, ) = 5(n +, + )
98] WEIGHTED STIRLING NUMBERS OF THE FIRST AND SECOND KIND II 249 (4.6) R (n 9 9 ) = S (n, ) 9 R (n 9, ) = S (n +, + ). In (2.9), tae X =. Then, by (.) (4.5) with replaced by n - 9 > *. J. o)( n 2v) - i > * - ^( n + 2i _ )- It follows that (4.7) B(fc, j, ) = B(, j + ), ( < j < ) ; B(,, ) =. Similarly, taing X = in (2.9), we get j-o Thus J-I (4.8) (, j, ) = B( 9 j ) s ( l < i < ) ; B(,, ) =. Net, tae X = in (.8), we get This gives j-o (4.9), B^, j, ) = B ( 9 j + ), ( j < ) ; ^ (,, ) =. Similarly, we find that (4.) B^ 9 J, ) = B ( s j ), ( <. j <. ); ^ (,, ) =. It is easily verified that (4.9) (4.) are in agreement with (4.4). Moreover, for X =, (4.) reduces to by (4.8) (4.9), this becomes.j-i B (, j, ) = B( 9 - j, ); B ( 9 j + ) = B( 9 - j ), which is correct. For X =, (4.) reduces to by (4.7) (4.), this becomes as epected. B ( 9 J, ) = B( 5 - j, ); B ( 9 j) = B( 9 - j + ) 5. THE COEFFICIENTS B(, j, X); B (, J, X) It is evident from the recurrences (2.4) (.) that (, J, X) B ( 9 j, X) are polynomials of degree < in X with integral coefficients. Moreover, they are related by (4.). Put (5.D f a 9 ) = Y. B ( > o> X^J =
25 WEIGHTED STIRLING NUMBERS OF THE FIRST AND SECOND KIND II [Oct. (5.2) / l f c (X, ) = B i ( > > *>* j - o By (4.), we have (5-) / l f (X, ) -%(l - X, ). By (2.7) (.6), In the net place, by (2.4), (5.) becomes f (X, a?) = { ( f c + J - X)B(fe -, J, X) Since + (& - J + X)B(fe -, J -, X)}a;. ] (fc + J - X)B(fc -, j, X)ar* = (fc - X +a;z?)/ - (X, a;) ( f e - J + X)B(fe -, j -, X)a?* = ^Qc - j - + X)B(fc -, j, X)a: j = o j = o where D = dld 9 i t follows that fc-l = a?(fc - + X - D)f _ (X 9 ) 9 (5.5) / fc (X, ) = { - X+ ( - + X)a? + (l - )B}f ^{\ 9 ). The corresponding formula for f (\ 9 ) i s (5.6) f lf (X, a?) = { - X + (2fc - 2 + X)a? + a;(l - ^ P } ^ ^. ( X, a? ). Let E denote the familiar operator defined by Ef{n) = f(n + ). Then, by (2.9) (5.), we have (5.7) R(n 9 n -, X) = / fc (X, E)(%\ Similarly, by (.8) (5.2), (5.8) R^n 9 n -, X) = / lffc (X, #)(j)- Thus, the recurrence R(n +, n - +, X) - R(n 9 n -, \) = (X+-n-+l)R(n 9 n - +, X) becomes Since we have A<*- ^(V/)-A^»" B >( 2 B fc) -<* +» - * + DA.i». ^( 2 fe w - 2)- /?2 + \ / n \ _ ( n \ \ 2 ) \2) " \2 - l) 9 (5.9) f {\ 9 E)( 2 n _ ^ = (X +n - + D/fc.^X, *)( 2 n _ 2 ). Applying the finite difference operator Zl we get (5.) 4 ( A, ^ ( ^ l i ) = (X + n-fe+2)4. ( X, *)( 2fe l ) + / ^ ( A, *)( 2fe n _ 2 )
98] WEIGHTED STIRLING NUMBERS OF THE FIRST AND SECOND KiND I I 25 Similarly, the recurrence i? (n +, n - + l s X) - R ± (n s n - s X) = (X + n)rln, n - +, X) yields (5.) / l ( X, E)( n 2 _ : ) = (X + n ) / l f _ t t, ^)( n 2fe _ 2) (5.2) f it a, E)[ n 2 _ U (X + n -f D / ^ ^ t t, ^ ) L n _ ) i,-i\2-2} 6. AN APPLICATION We shall prove the following two formulas: (6.) R(n, n -, - X) = ( V - V ) ( \ ~ + V ) f l i < f e + * *» A > (6.2) fl^n, n - ft, - X) = X ( \ + - V (^fc + V ) * ( / c + * * 9 X ) Note that the coefficients on the right of (6.) (6.2) are the same. To begin with, we invert (2.9) (.8). It follows from (2.9) that 2>(n, n -, \) n -* = ><*, j, \) -J (" 2 VV" 2 * + j ^>" = n m = n \ so that ^B(fe, - j, X)^ = ( - ^ ) 2 + X i? ( n s n " ^ ^ ) * n " * j - n = & = L(-iW 2 VVE *<* + * *> «i _ n \ ^ + _ n It follows that (6.) B(fc, - j, X) = ^ ( - ^ " ( T ^ t V ^ + ts *» X) S i m i l a r l y, J = (6.4) BA - fc - j, X) * - o ^ c By (2.9), (4.), (6.4), we have B(n, n - s - X) - X V ^ fe " ^> A >( n 2fc J ) j - o v / t(" 2 V)i < - ) "t?- + t h«+ t - 4 J = /fo X
252 WEIGHTED STIRLING NUMBERS OF THE FIRST AND SECOND KIND II [Oct. (6.5) t - o The inner sum is equal to L j = t E(^ev)(»M-)"(" 2 r)e^ -t - (~2 - ). (n + t + ). (-fc + t). t _ 2 + l) d (-fc + ty. (n + t\ ^2-2 -, n + t +, -Zc + t" The F is Saalschiltzian [, p. 9] 9 we find 9 after some manipulation that es Thus 9 (6.5) becomes E n./2fe + l\/n + + j \ /fc + n + l\/fc - n - l\. = ( " } V j A 2 ) = { ~ t ) \ + t ) R(n 9 n This proves (6.). The proof of (6*2) is eactly the same. of 7. BERNOULLI POLYNOMIALS OF HIGHER ORDER Norlund [9, Ch. 6] defined the Bernoulli function of order z by means (7.) <*V It follows from (7.) that B n (A) is a polynomial of degree n in each of the parameters s 9 A. Consider (7.2) Kn, n -, A) = (^B ( -" + «(X) (7.) (n, «- fc, A) = ( fe " i y ^ n + ) ( l - A ). It follows from (7.2) that E oo.. *. # - E( n * ><-(A)^ - & **>(*>. Hences by (7.) 9 we have (7.4) ^fi(nj, «J = i( e»-l)v». n = Comparison of (7.4) with (.7) gives Q(n, s X) = R(n 9 9 X), so that (7.5) i?(w, n - 9 X) = ^ ) 5 ( " n + f c ) ( A ).
98] WEIGHTED STIRLING NUMBERS OF THE FIRST AND SECOND KIND II 25 Net, by (7.), " n= - j n n= = n = It is nown [8, p. 4] that U K l Thus, t n U + V) "-LOgU + V)) = 2^-rn = (i + w-hiogd + *»* = E otttyrerw = n = fc = * " X Ai/ = ( - u ) " A ( l - M ) " *. T h e r e f o r e, Q (n, fc 9 X) = R (n 9 9 X), so t h a t (7.6) ^ ( n, n - fc, X) = (^ " * " ^ " ^ ( l - A). For X =, (7.5) reduces to S(n, n-) = [l)bi- n+) ; for X =, we get sin + i. - * + i) = (^r^cd = (^)(i -Z^TTT^TK^-" For X = l s (7.6) reduces to 5 l ( + i, n-fc + ) = ( f e - ^ ~ ^ B ^ " ; for X =, we get *i<».»-«-(*r )( i - K = f r K Thus, in all four special cases, (7.5) (7.6) are in agreement with the corresponding formulas for S(n 9 n - ) S (n 9 n - ). We may put 8. THE FUNCTIONS 2?(n,, A) AND ^(n, fc, X) (8.) R(n 5 n - fc, A) = i?(fc, j, *)( 2fe W _ j) (8.2) fl^n, n - fe, X) - ] [ > ( &, j, X)( W 2fe _ ^. j-o
25^ WEIGHTED STIRLING NUMBERS OF THE FIRST AND SECOND KIND II [Oct. The upper limit of summation is justified by (2.9) (.8). Using the recurrence (2.), we get Since R(n +, n - +, X) - R(n 9 n - 9 X) fc-i., = (n - + + A)]p?(fc -, j, X)L fe J. _ A -l = ( 2 Z c - - l)r( -, J, X)( 2 f e _ " n. _ ^ \2 - j fe-i, > + (fc ~ J - + )R( -,, )y 2 J. _ \. R(n +, n - +, X) - R(n, n - 9 X) we get (8.) 2?(fc, j, X) = (2- j - l)r(-, j, X)+ (fc- j + X)i?(fe-, j -, X), For =, (8.) gives (8.4) i? (, 9 X) =, i? (, j, X) =, (j > ). The following values are easily computed using the recurrence (8.). R(, j, X) K N ^ 2 4 i X 2 + X X 2 5 + 5X + 4X + 6X 2 X 4 5 5 + 5X 25 + 6X + 45X 2 + 5X + X 2 + 4X X h (8.5) (8.6) Also, It is easily proved, using (8.), that R f ( 9, X) =..5... {I - ) R f ( 9 9 X) = X \ (8.7) ](-l) J VO: s j, X) = ( - X ) < Moreover, it is clear that R f ( 9 j, X) is a polynomial in X of degree To invert (8.), multiply both sides by (-) f J sum over n. Changing the notation slightly, we get * (8.8) R(, - j, ) = )<-i> i+ *( + i) R ( + * *> >-
98] WEIGHTED STIRLING NUMBERS OF THE FIRST AND SECOND KIND IS Turning net to (8.2) employing (.2), we get R (n +, n - +, X) - R (n s n - 9 X) fc-l = (n + \)J2 R {( ~. > W[ 2 -"a - 2) " = X ^ - ~ DRli - l, j\ \)[ 2 _ n. _ A + ] (2fe - J - 2 + X)R[( -, j, X ) ( ^ _*. _ 2 j. I t follows that (8.9) R[( 9 j, \)=(2-j-l)R f (-l 5 j, X) + (2fe- j - l+a)i? f (/c-, j -, X ), For fc =, we have (8.) R±(fl 9, X) =, Rl(, j \ X) =, (j > ). The following values are readily computed by means of (8.9) (8.). N\ RIOt, J Y X ) 2 4 X 2 2 + X (X) 2 4 5 5 2 + 5X 2 + 5X 6 + 4X + 6X 2 + 65X + 45X 2 < * > 24 + 7X + 5X 2 + X an (8.) (8.2) Also (8.) We have R{(,, X) =..5... (2 - ) R(( s 9 X) = (X) fc. ^ ( - D ^ t f c, j\ X) = ( - X ). J - Clearly 9 R[(, j, X) is a polynomial in X of degree j. Parallel to (8.8), we have (8.4) R{(> - j, X) = Yl ("D J + t ft + jk(fc + Y *> A).
256 WEIGHTED STIRLING NUMBERS OF THE FIRST AND SECOND KIND I I [Oct. 9. ADDITIONAL RELATIONS (Compare [, 4].) By (8.4) (.), we have R[{ 9 - j, X) = (-l)*( \ ^R^ + j - t, j - t, X) /, A & 8= = t= t = It can be verified that the inner sum is equal to ( - / X.. Thus, / / \ (9.) R[(, j, X) = X! u F i ( / c > s > * ) Similarly, (9.2) i?(fc, fe - j, «= (- f. W, s, X). The inverse formulas are (9.) B (, t, X) = ( - D r f ~ * ( j W ( f e, J X ) ^"* (9.4) B(, t, X) - Y,(-l) i t ll\ri, J\ X). j - * In t h e net p l a c e, by (9.4) (. ), R^n, n -, X) = X B i ^ * t = o X >("2fe*) = Z 5 ( f c > * - *.! - > ( " 2fe *) / t = V / - *<*.*. i - ) (» + - y = / The inner sum is equal to (-l)" 7 ) j» = \ / \ / V 2 - I \, therefore (9.5) R (.n, n -, X) = ( - l ) * ^ * jv?(fc. * - J. - X). S i m i l a r l y, y - (9.6) i?(n, n -, X) = (-D fc " J f^ + ffii^9 " J 5 X " X) : The inverse formulas are less simple. We find that
98] WEIGHTED STIRLING NUMBERS OF THE FIRST AND SECOND KIND II 257 (9.7) R{(n,, A) = Y^{~l) n -C n {, j)r(n + j, g, - X) j-o (9.8) R(n,, X) = (-D n " J^n(^5 J)^i(w + j, j, - X), where j-o It does not seem possible to simplify C n ( 9 j). We omit the proof of (9.7) (9.8). Finally 9 we state the pair (9.) R{(n s, X) = ( - D * ( Z f} Rf (ri 9 t, - X),. _ i (9.) ;?<n, fe, X) = ( - l ) * (._ J W ( n, *, - X). The proof is lie the proof of (8.8) (8.4). REFERENCES. W. N. Bailey. Generalized Eypergeometric Series. Cambridge, 95. 2. L. Carlitz. "Note on Norlunds Polynomial 5. s)." Proa. Amer. Math. Soc. (96):452-455.. L. Carlitz. "Note on the Numbers of Jordan Ward." Due Math. J. 8 (97):78-79. 4. L. Carlitz. "Some Numbers Related to the Stirling Numbers of the First Second Kind." Publications de la Faculte d f Electrotechnique de l! Vniversite a Belgrade (977)^49-55. 5. L. Carlitz. "Polynomial Representations Compositions, I." Houston J. Math. 2 (976):2-48. 6. L. Carlitz. "Weighted Stirling Numbers of the First Second Kind I." The Fibonacci Quarterly 2 (98):47-62. 7. G. H. Gould. "Stirling Number Representation Problems." Proc. Amev. Math. Soc. (96):447-45. 8. L. M. Milne-Thomson. Calculus of Finite Differences. London: Macmillan, 95. 9. N. E. Norlund. Vorlesungen uber Differen&enrechnung. Berlin: Springer, 924.