4420 Legendrian and Bessel Functions Polymorphic Functions 4430 



J (k) and J T (k) (or 1 and 1^) from k = to 



A=15-5, table. Meissel, E. Berl. Ak. Ab. 



(1888) (Anh. Mth.) 23 pp. 

 I functions, expansions. Meissel, E. As. Nr. 



127 (1891) 359-. 

 ij', theory. MmseZ, . As. Nr. 128 (1891) 



145-. 

 J n -(x), new formulae. Kapteyn, W. Bll. Sc. 



Mth. 16 (1892) 41-. 

 ^ w +i( x ) = ( integral), place of roots. 



Rudski, M. P. Liege S. Sc. Mm. 18 (1895) 



No. 3, 29 pp. 

 O n (x\ and S n (x). Crelier, L. C. E. 125 



(1897) 421-, 860-. 



(1 - '2aH + a 2 ) * , expansion in powers of a. 



Hansen, P. A. Leip. Ab. Mth. Ps. 1 (1852) 



123-. 

 ^{r-2 + r '2 _ 2rr' (cost/cost/' + sinf/sin U'cosJ) } , 



expansion of negative odd powers. Hansen, 



P. A. [1854] Leip. Ab. Mth. Ps. 2 (1855) 



283-. 



Schlomilch Z. 3 (1858) 244-. 



(1 - 2qx + q 2 ) ~, and {1 - 2q [cos cos 0' 



+ sin sin 0' cos (\f/ - 1 



expansion. 4M#, (Dr) Jkf. Wien Ak. Sb. 51 

 (1865) (Ab. 2) 429-. 



^f ,' ^ J)i theory. Thomae, J. [1867] 



Z a Mth. Ps. 14 (1869) 48-. 

 E n . Hermite, C. A. Mt. 3 (1869-70) 83. 

 Y m and X m . Gegenbauer, L. Wien Sb. 65 



"(1872) (Ab. 2) 373-; 66 (1872) (Ab. 2) 55-; 



68 (1873) (Ab. 2) 357-. 

 , integral expressions for. Gegenbauer, L. 



Wien Sb. 66 (1872) (Ab. 2) 374-. 

 X~ r+1 , expansion in terms of. Gegenbauer, L. 



Wien Sb. 66 (1872) (Ab. 2) 415-. 

 tJl-2-riU+i) 1 , expansion of negative odd 



powers. Backlund, J. 0. [1877] St Pe"t. 



Ac. Sc. Bll. 24 (1878) 509-. 

 G v n (x). Gegenbauer, L. Wien Ak. Sb. 75 (1877) 



\Ab. 2) 891-. 

 Y n , final value for infinitely increasing values 



of n. Escary, . As. Fr. C. E. 8 (1879) 273-. 

 , expansions in series whose terms are these 



functions. St Germain, A. de. C. B. 88 



(1879) 1186-, 1313-. 

 P(cos7), n infinite. Heine, H. E. [1880] 



Crelle J. Mth. 90 (1881) 329-. 

 P n (0037), development. Pleskot, A. Prag Sb. 



(1893) (Mth.-Nt.) No. 17, 7 pp.; Fschr. 



Mth. (1893-94) 820. 



J -I ^ l> J_! tC ' '' J_i J 



/ P i sin mOdn. Forsyth, A. R. QJ. Mth. 

 17 (1881) 37-. 

 C v n (x) where (1 - 2zx + a; 2 )-" ="2" c" n (x) z n . 



Gegenbauer, L. Wien Ak. D. 48 (1884) 



(Ab. 2) 293-. 

 , theorems. Gegenbauer, L. Wien Ak. D. 



57 (1890) 425-. 

 , on roots. [Coefficient of z 2 in expansion 



of (l-2zx + z z )~ v .] Gegenbauer, L. Amst. 



Ak. Vs. 8 (1900) 250- ; Amst. Ak. P. 2 (1900) 



196-. 

 Y m (x), addition theorem. Gegenbauer, L. 



Wien Ak. Sb. 92 (1886) (Ab. 2) 1340-. 

 T(x). Gegenbauer, L. Wien Ak. Sb. 95 



"(1887) (Ab. 2) 274-. 

 X n =0, roots. Stieltjes, T. J. Acta Mth. 9 



(1887) 385-. 



Gubler, 



E. Mth. A. 49 (1897) 583- 



Y and y lt table. Smith, B. A. Mess. Mth. 

 26 (1897) 98-. 



G (x) , G 1 (x) , and J n (x *Ji) , numerical com- 

 putation. Aldis, W. S. E. S. P. 66 (1900) 

 32-. 



4430 Polymorphic functions. Other 

 functions which may be denned 

 by definite integrals. (See also 

 4860.) 



Abelian generating functions. Minine, A. 



Les Mondes 2 (1882) 126-. 

 . Pareto, V. Crelle J. Mth. 110 



(1892) 290-. 

 Associated functions and potential of spherical 



segment. Beltrami, E. N. Cim. 14 (*1883) 



139-. 



polynomials ( | U^ *V V , idx = 0\ . Didon, 



F. Par. EC. Norm. A. 6 (1869) 111-. 

 Conical functions, adjunctive, properties of 



integrals. Leonhardt, G. Mth. A. 19 (1882) 

 578-. 

 Definite integrals to express the roots of 



u m -pu n -q = 0. 



Nekrasov, P. A. Eec. Mth. (Moscou) 13 

 (1886) 739-; Fschr. Mth. (1888) 304. 



, functions involved under. Gomes de 

 Souza, J. E. S. P. 8 (1856-57) 146-. 



Double gamma function. Barnes, E. W. [1900] 

 Phil. Trans. (A) 196 (1901) 265-. 



functions, genesis. Barnes, E. W. 

 L. Mth. S. P. 31 (1900) 358-. 



Formula of Anger for 



1 2>r cos (ha. - k sin a) da = 2ir ij 



and analogous formulae. Cauchy, A. L. 



C. B. 39 (1854) 129-. 

 Functions analogous to Laplace's. Routh, E. J. 



L. Mth. S. P. 11 (1879-80) 92-. 

 Legendre's. Aleksyeev, N. N. (xn) 



Eec. Mth. (Moscou) 5 (1870) (Pt. 1) 125-. 

 , arbitrary, expressed by double integrals. 



Pioch, A. Brux. Mm. Cour. 4, 15 (1841-42) 



74pp. 



349 



