Hi thanks for your question.
Your 84 years old mother is having pain left shoulder and left arm many months after a traumatic left hemiplegia
.She is 84 years old and most probably this pain has some other origin and not related to her old trauma.The commonest cause of pain joint at this age is osteoporsis(decrease bone masss due to loss of calcium) and osteo arthritis
(inflammation of joints).I would suggest you to consult a orthopediac specialist who may be able to evaluate your mother for this pain and accordingly he/ she may treat her pain.She may need some blood and radiological work up.
Hope this answers your question.