The Brazilian's absence is a significant setback, with the forward having scored seven goals and provided three assists in the Premier League this season.
"Richy unfortunately got a hamstring injury which will keep him out for up to seven weeks," Frank told reporters.
Richarlison's injury comes as Spurs look to bounce back from their FA Cup third-round exit at home to Aston Villa.
Their Danish manager stressed that his team should focus on their own performance rather than on the fact that it is a derby.
"I know it's a big London derby, of course, against West Ham, with the rivalry and everything, but it's all about us. It's how we come out with energy, positive, forward," he said.
Frank added that Ivory Coast midfielder Yves Bissouma is set to return after his country lost in the quarter-finals, while Lucas Bergvall might be available. Full back Destiny Udogie and striker Dominic Solanke are also closer to full fitness.
Captain Cristian Romero will return to the squad after serving a one-match suspension.
Spurs are 14th in the standings with 27 points from 21 games, while West Ham are languishing in the relegation places with 14 - seven points from the safety zone.
Frank also praised Tottenham's capture of midfielder Conor Gallagher from Atletico Madrid, calling it a "top signing" and highlighting the 25-year-old former Chelsea player's Premier League experience and leadership qualities.
"First and foremost, he is a very good player, great experience in the Premier League, also in La Liga, captained teams. So that character, that leadership, that experience, and still young," Frank said.
