Here is a colorful rejoinder by D. Zagier (in his reprinted article on the dilogarithm) to colorful language by Ph. Elbaz-Vincent and H. Gangl:
[Ph. Elbaz-Vincent and H. Gangl] called these functions "polyanalogs," an amalgam of the words "analogue," "polylog," and "pollyanna" (an American term suggesting exaggerated or unwarranted optimism). Presumably the correct term for the case $m=2$ would then be "dianalog," which has a pleasing British flavo(u)r.