There are many lith developer formulas available, both the traditional AB type and others. Most do not contain formaldehyde.

Ansco 81 Long Life Reprolith Developer
For lithographic film.
Water (52C) 750 ml
Hydroquinone 35 g
Sodium sulfite (anh) 55 g
Sodium carbonate (mono) 80 g
Citric acid 5.5 g
Potassium bromide 10 g
WTM 1 l
Do not dilute. Develop about 3 minutes at 20C.

DuPont 15-D high contrast (litho) developer
Solution A
Water 750 ml
Hydroquinone 45 g
Sodium sulfite (anh) 30 g
Sulfuric acid (conc) 4 ml
WTM 1 l
Solution B
Water 500 ml
Sodium carbonate (anh) 30 g
Potassium carbonate 90 g
Potassium bromide 8.3 g
Sodium sulfite (anh) 90 g
WTM 1 l
Mix 1:1 for use and develop 4 to 5 minutes.

GAF 81 process film developer
Water (52C) 750 ml
Hydroquinone 35 g
Sodium sulfite 55 g
Sodium carbonate 80 g
Citric acid 5.5 g
Potassium bromide 10 g
WTM 1 l
Develop lithographic film about 3 minutes at 20C.
Ansco/Agfa/GAF 81 has been around for a long time, and there are similar Kodak and Ilford recipes as well those that substitute potassium or sodium metabisulfite for the Citric Acid. GAF 81 is the only one to claim a long working life, however.
Variations (suggested for use as a lith print developer):
Substitute Catechol for Hydroquinone in your formula. Catechol is more active than Hydroquinone with about the same level of human toxicity (low, in solution).
Ascorbic Acid is a possibly useful additive/accelerant. It is also an oxygen scavenger and may be used to replace some of the sulfite. It does not, however, produce infectious development.

Kodak D-9 hydroquinone caustic film developer
For extreme contrast with process films
Solution A
Water (53C) 500 ml
Sodium bisulfite 22.5 g
Hydroquinone 22.5 g
Potassium bromide 22.5 g
WTM 1 l
Solution B
Water 1 l
Sodium hydroxide 52.5 g
Use equal parts of A and B. Wash negative thoroughly after development and before fixing to avoid stains. Use at 18C.