NOTE ON MODULAR INVARIANTS. By Dr. L. E. DICKSON. sem elementary standpoint, the functions. -། or more varies which are invariant under rus cuate, modulo p (a prime), of deter sar is de case of a binary form ..... we get c=(-1)'c.. Hence c=(-1) c exists no invariant binary form. Varg ne transormation (+ey, y) to (1), we get A ད、 ང་ we videor the transformation with e=1. Ten C=cm=0 shows tha ewus *\____]+p>2* C, ..., C2, give To prove by induction that, if p>3, Cp., ..., Cap-3 give CH=0, C 0, Cap-3 = 0, p+2 Cp+1=0,..., Cpp-3=0 for 1 <p<p-1. Then C., gives (5) let c, The second and third binomial coefficients are evidently multiples of p; also the first is a multiple of p since m is a multiple of p and pp is not. Hence We shall prove by a two-fold induction that (6) C(-1)p11=0, C(j-1)p+2=0, •••9 Cjp-j-1=0 (j=1, 2, ..., m/p). We assume that these relations hold for j=1,..., s and prove that they hold also for j=s+1. They are true for j=1 and j=2 by (3) and (4). By our hypothesis c, (isp) is zero unless has one of the values (7) 0, jp-j, jp-j+1, ..., jp (j=1, ..., 8). We proceed to prove that Cp. C Cl- give viz., relations (6) for j=8+1. We may set p>8+2, since in the contrary case the statement is vacuous. proof by induction, let To make this Cap=0, Cp=0,..., Cp.P-=0 (1<p<p-s). Hence in C, for r=sp+p, the only c, not known to be zero is that with sp+p-1 and those for which i has one of the values (7). The coefficient (") of c, is a multiple of numbers (7), other than 0, are i=jp-k (0 ≤k<j≤8). Since m is of the form pu, the coefficient of c, in C, is which is a multiple of p. The p. Indeed, (4) is a multiple* of p iť a = £a,p',_b = 2b,p* (0 ≤a,<p, 0 ≤b;<p), * Dickson, Annals of Mathematics, vol. xi. (1896), p 75. DERIVED RELATIONS INVOLVING B, En, §§ 18-23. V-formulæ ..., §§ 18-27. Relations between Bernoullian numbers, derived from the 94 S$ 24-27. Relations involving En, In, Jn, derived from the V-formulæ 99 §§ 28-33. §§ 31-35. GENERAL AND SPECIAL U-FORMULE (NOT PROCEEDING BY POWERS), §§ 28-35. General formulæ relating to Un (x) 102 Special cases of the formulæ relating to Un(x) 105 $36. $$ 37-38. DERIVED RELATIONS INVOLVING Bn, En, $$ 36-41. 106 Relations involving Bernoullian numbers derived from the U-formulæ 106 §§ 39-41. Relations involving En, Hn, Pn, Qa, Tn derived from the U-formulæ 107 GENERAL AND SPECIAL - AND U-FORMULE PROCEEDING GENERAL AND SPECIAL V- AND U-FORMULE PROCEEDING RELATIONS INVOLVING Ba, En, AND PROCEEDING SPECIAL RELATIONS INVOLVING Bn, En, CORRESPONDING TO 1, 2, 3, . p. 43, 1. 5, for (i) read (iii). p. 48, 1. 4 and 1. 8, and p. 49, 1. 4, for he read han p. 49, 1. 15, for 3 read =33. p. 62, 1. 15, for Pa-2 read Pn-1 p. 75, 1. 6, for 32′′ – 2 read 32′′ – 3. p. 89, 1. 4, for § 147 read § 145. p. 94, 1. 17, for n=2n read n=2m. p. 102, 1. 1, for § 178 read § 177. p. 102. The formula in §194 is not true except for c=1, and this section should be omitted. The error arises from writing - instead of eac-1 in the denominator in 1. 7 of the section. The formula in the case of c=1 is at once derivable from the equation on p. 102, 1. 2, by putting æ+1, x+2, x+-1 for x and adding. NOTE ON MODULAR INVARIANTS. By Dr. L. E. DICKSON. WE consider, from an elementary standpoint, the functions of two or more variables which are invariant under every linear transformation, modulo p (a prime), of determinant unity. Consider first the case of a binary form The conditions for invariance of B under (y, x) are c=(-1) c Cm-i (i=0, 1, ..., m). cm-i Replacing i by m-i, we get c=(-1)'c,. Hence c=(-1)" c.. If m is odd and p>2, there exists no invariant binary form. Applying the transformation (x+ey, y) to (1), we get which is identical with (1) if, and only if, For the present we shall employ the transformation with e=1. We first assume that c0. Then C=cm=0 shows tha m is a multiple of the modulus p. If p>2, C,..., C, give To give a proof by induction, we assume that, for 1<r<p, c=0,..., C=0. Then C, gives The first binomial coefficient is a multiple of p and the second is not. Hence c,, = 0. Next, C, gives P+1 Now C is satisfied if p>2, and is considered under C-1 if p = 2. *The case p = 2 is not exceptional; (3) is then a true but vacuous statement. Similarly in (5) if p 3. |